6.4: Energy and Metabolism - Metabolism of Carbohydrates
- Analyze the importance of carbohydrate metabolism to energy production
Metabolism of Carbohydrates
Carbohydrates are one of the major forms of energy for animals and plants. Plants build carbohydrates using light energy from the sun (during the process of photosynthesis), while animals eat plants or other animals to obtain carbohydrates. Plants store carbohydrates in long polysaccharides chains called starch, while animals store carbohydrates as the molecule glycogen. These large polysaccharides contain many chemical bonds and therefore store a lot of chemical energy. When these molecules are broken down during metabolism, the energy in the chemical bonds is released and can be harnessed for cellular processes.
Energy Production from Carbohydrates (Cellular Respiration )
The metabolism of any monosaccharide (simple sugar) can produce energy for the cell to use. Excess carbohydrates are stored as starch in plants and as glycogen in animals, ready for metabolism if the energy demands of the organism suddenly increase. When those energy demands increase, carbohydrates are broken down into constituent monosaccharides, which are then distributed to all the living cells of an organism. Glucose (C 6 H 12 O 6 ) is a common example of the monosaccharides used for energy production.
Inside the cell, each sugar molecule is broken down through a complex series of chemical reactions. As chemical energy is released from the bonds in the monosaccharide, it is harnessed to synthesize high-energy adenosine triphosphate (ATP) molecules. ATP is the primary energy currency of all cells. Just as the dollar is used as currency to buy goods, cells use molecules of ATP to perform immediate work and power chemical reactions.
The breakdown of glucose during metabolism is call cellular respiration can be described by the equation:
C 6 H 12 O 6 +6O 2 →6CO 2 +6H 2 O+energy
Producing Carbohydrates (Photosynthesis)
Plants and some other types of organisms produce carbohydrates through the process called photosynthesis. During photosynthesis, plants convert light energy into chemical energy by building carbon dioxide gas molecules (CO 2 ) into sugar molecules like glucose. Because this process involves building bonds to synthesize a large molecule, it requires an input of energy (light) to proceed. The synthesis of glucose by photosynthesis is described by this equation (notice that it is the reverse of the previous equation):
6CO 2 +6H 2 O+energy→ C 6 H 12 O 6 +6O 2
As part of plants’ chemical processes, glucose molecules can be combined with and converted into other types of sugars. In plants, glucose is stored in the form of starch, which can be broken down back into glucose via cellular respiration in order to supply ATP.
Contributions and Attributions
- OpenStax College, Energy and Metabolism. October 26, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest/ . License : CC BY: Attribution
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44421/latest...ol11448/latest . License : CC BY: Attribution
- energy. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/energy . License : CC BY-SA: Attribution-ShareAlike
- metabolism. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/metabolism . License : CC BY-SA: Attribution-ShareAlike
- bioenergetics. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/bioenergetics . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Energy and Metabolism. October 26, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest/ . License : CC BY: Attribution
- OpenStax College, Introduction. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44421/latest...e_06_00_01.jpg . License : CC BY: Attribution
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44425/latest...ol11448/latest . License : CC BY: Attribution
- kinetic energy. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/kinetic%20energy . License : CC BY-SA: Attribution-ShareAlike
- potential energy. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/potential%20energy . License : CC BY-SA: Attribution-ShareAlike
- Energy. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Energy . License : CC BY-SA: Attribution-ShareAlike
- chemical energy. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/chemical_energy . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Energy and Metabolism. October 26, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest/ . License : CC BY: Attribution
- OpenStax College, Introduction. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44421/latest...e_06_00_01.jpg . License : CC BY: Attribution
- OpenStax College, Potential, Kinetic, Free, and Activation Energy. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44425/latest...06_03_01ab.jpg . License : CC BY: Attribution
- OpenStax College, Potential, Kinetic, Free, and Activation Energy. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44425/latest...06_03_02ab.jpg . License : CC BY: Attribution
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest...ol11448/latest . License : CC BY: Attribution
- catabolism. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/catabolism . License : CC BY-SA: Attribution-ShareAlike
- enzyme. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/enzyme . License : CC BY-SA: Attribution-ShareAlike
- anabolism. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/anabolism . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Energy and Metabolism. October 26, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest/ . License : CC BY: Attribution
- OpenStax College, Introduction. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44421/latest...e_06_00_01.jpg . License : CC BY: Attribution
- OpenStax College, Potential, Kinetic, Free, and Activation Energy. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44425/latest...06_03_01ab.jpg . License : CC BY: Attribution
- OpenStax College, Potential, Kinetic, Free, and Activation Energy. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44425/latest...06_03_02ab.jpg . License : CC BY: Attribution
- OpenStax College, Energy and Metabolism. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest...e_06_01_03.jpg . License : CC BY: Attribution
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest...ol11448/latest . License : CC BY: Attribution
- NADPH. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/NADPH . License : CC BY-SA: Attribution-ShareAlike
- glucose. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/glucose . License : CC BY-SA: Attribution-ShareAlike
- adenosine triphosphate. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/adenosine%20triphosphate . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Energy and Metabolism. October 26, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest/ . License : CC BY: Attribution
- OpenStax College, Introduction. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44421/latest...e_06_00_01.jpg . License : CC BY: Attribution
- OpenStax College, Potential, Kinetic, Free, and Activation Energy. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44425/latest...06_03_01ab.jpg . License : CC BY: Attribution
- OpenStax College, Potential, Kinetic, Free, and Activation Energy. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44425/latest...06_03_02ab.jpg . License : CC BY: Attribution
- OpenStax College, Energy and Metabolism. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest...e_06_01_03.jpg . License : CC BY: Attribution
- OpenStax College, Energy and Metabolism. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44422/latest...e_06_01_02.jpg . License : CC BY: Attribution
Key Points
- The breakdown of glucose living organisms utilize to produce energy is described by the equation: C 6 H 12 O 6 +6O 2 →6CO 2 +6H 2 O+energy.
- The photosynthetic process plants utilize to synthesize glucose is described by the equation:6CO 2 +6H 2 O+energy→ C 6 H 12 O 6 +6O 2
- Glucose that is consumed is used to make energy in the form of ATP, which is used to perform work and power chemical reactions in the cell.
- During photosynthesis, plants convert light energy into chemical energy that is used to build molecules of glucose.
Key Terms
- adenosine triphosphate : a multifunctional nucleoside triphosphate used in cells as a coenzyme, often called the “molecular unit of energy currency” in intracellular energy transfer
- glucose : a simple monosaccharide (sugar) with a molecular formula of C6H12O6; it is a principal source of energy for cellular metabolism


32.2: Anorexiants
By the end of this section, you should be able to:
- 32.2.1 Identify the characteristics of anorexiant drugs used for weight management.
- 32.2.2 Explain the indications, actions, adverse reactions, and interactions of anorexiant drugs used for weight management.
- 32.2.3 Describe nursing implications of anorexiant drugs used for weight management.
- 32.2.4 Explain the client education related to anorexiant drugs used for weight management.
Anorexiants are drugs used to promote weight loss through appetite suppressant and stimulation effects by increasing norepinephrine availability to neural receptors. Although these drugs have gone through the standard U.S. Food and Drug Administration (FDA) approval process, many have been removed from the market due to the critical and fatal effects of body stimulation.
Phentermine Hydrochloride
Phentermine hydrochloride is a sympathetic amine that stimulates the central nervous system to increase norepinephrine (NE) availability. When NE is attached to brain receptors, it suppresses the appetite and increases metabolism. Along with following a healthy diet, phentermine hydrochloride is used adjunctly as an oral tablet or capsule to promote weight loss. It is recommended for short-term therapy (8–12 weeks) as a daily dose or three times a day before meals.
Phentermine hydrochloride is a Schedule IV controlled substance because there may be a potential for problematic use; it should not be used for clients with a history of substance misuse or substance use disorder.
Phentermine and Topiramate ER
The drugs phentermine (an anorexiant and stimulant) and extended-release topiramate (an anticonvulsant and gamma-aminobutyric acid [GABA]) are combined in a capsule for weight management. It is utilized in addition to daily caloric restriction and increased physical activity in clients with an initial BMI of 30 kg/m 2 or greater and at least one other comorbidity of hypertension, type 2 diabetes mellitus, and/or dyslipidemia.
Clients are initially started on a low dose of the drugs for 14 days; the dose is then increased yet still taken once per day for 12 more weeks. Weight is assessed after 12 weeks of the increased dosage. If weight loss has not exceeded 3% of the baseline weight, the dose is either discontinued or increased again, with an increased dosing for another 14 days and increased again for 14 weeks. Baseline weight at that point is reassessed, looking for a 5% or more weight loss over baseline. Daily doses are not increased in clients with renal and/or hepatic insufficiency.
Phentermine and topiramate ER is also a Schedule IV controlled substance.
Phendimetrazine
Phendimetrazine is an appetite suppressant used short-term to promote weight loss in conjunction with a low-calorie diet and exercise in clients who did not lose weight with diet and exercise alone. The drug is available in immediate-release (IR) tablets, extended-release (ER) capsules, and sustained-release (SR) capsules. It is important to swallow the extended-release capsule whole. Clients should not crush, break, or chew it.
Dosage varies for each client. Clients should be educated to follow the prescribed dosage and instructions. Dosing is determined by the provider based on both the client’s individual needs and the dosage form (Mayo Clinic, 2023a).
Nurses should advise clients that if they miss a dose, they should take it as soon as possible; however, if it is close to the time for the next dose, they should skip the dose. Clients should not take double doses. This medication is a stimulant and should never be increased without the provider's advice.
Benzphetamine
The drug benzphetamine is another CNS stimulant used as an anorexiant for weight loss in clients with obesity. Benzphetamine is an indirect sympathomimetic amine that acts like amphetamine, yet with fewer side effects. Its effectiveness as an appetite suppressant is thought to be due to the stimulating effects on the hypothalamus that release catecholamines. It is a Schedule III controlled substance for its potential for misuse and should not be used in clients with addictive behaviors.
Table 32.1 lists common anorexiants and typical routes and dosing for adult clients.
| Drug | Routes and Dosage Ranges |
|---|---|
|
Phentermine
( Adipex-P , Lomaira ) |
15–37.5 mg orally every morning.
Lomaira only: 8 mg 3 times daily. Administer 30 minutes before meals or 1–2 hours after meals. |
|
Phentermine and topiramate ER
( Qsymia ) |
Initial dose: 37.5 mg/23 mg orally for 14 days, then increase to 7.5 mg/46 mg daily. Weight is assessed after 12 weeks for dosage adjustment based on weight loss. |
| Phendimetrazine |
150 mg ER or SR capsule:
Orally once daily 30–60 minutes before morning meal.
35 mg IR tablet: Orally 2–3 times daily 60 minutes before meals; maximum dose: 70 mg/day. |
| Benzphetamine | 25–50 mg orally 1–3 times daily; best in 1 single mid-day dose; do not administer within 6 hours before bedtime to avoid insomnia. |
Adverse Effects and Contraindications
All four drugs discussed in this section should not be administered within 14 days of taking a monoamine oxidase inhibitor (MAOI), such as phenelzine, selegiline, isocarboxazid, or tranylcypromine. Using these medicines together may cause serious unwanted effects (Mayo Clinic, 2023b).
Use of phentermine hydrochloride is contraindicated in clients with hypersensitivity to sympathetic amines. The drug is a CNS stimulant, so it is often contraindicated in clients with hyperthyroid conditions, glaucoma, agitation, severe renal impairment, and/or cardiovascular disease such as coronary artery disease, stroke, arrhythmias, congestive heart failure, and uncontrolled hypertension (DailyMed, Adipex-P , 2020). Phentermine hydrochloride is used cautiously in clients with seizures, hypertension, diabetes mellitus, and renal impairment. Caution also should be exercised in clients with a history of substance use disorder because there is a risk of misuse with Schedule III and Schedule IV drugs.
Adverse effects of phentermine are related to the stimulant effects of the drug and include palpitations, tachycardia, arrythmias, precordial pain, nervousness, restlessness, dizziness, insomnia, anxiety, agitation, and tremors. Other adverse effects include hyper- or hypotension, syncope, pulmonary hypertension, fatigue, malaise, confusion, incoordination, headache, change in libido, gynecomastia, and hair loss. Severe hematological adverse effects are bone marrow suppression, agranulocytosis, and leukopenia. It is important to instruct clients not to take the drug late in the evening because it may cause insomnia.
Along with MAOIs, phentermine is contraindicated with tricyclic antidepressants, selective serotonin reuptake inhibitors (SSRIs), and antihypertensive drugs.
Life-threatening adverse drug reactions (ADRs) for phentermine and topiramate include hepatotoxicity and skin conditions such as Stevens–Johnson syndrome, erythema multiforme, and toxic epidermal necrolysis. Life-threatening seizures may occur following abrupt discontinuation of the drug. Clients should be monitored closely for changes in behavior, such as the development of or worsening depression and/or suicidal ideation.
Phentermine and topiramate ER combination is contraindicated in clients with hypersensitivity to either drug, sympathomimetics, or stimulants. It is also contraindicated in clients with glaucoma, hyperthyroidism, severe renal impairment, severe hepatic impairment, pregnancy, lactation, or history of suicidal ideation. Cautious use is recommended in clients with diabetes mellitus because there is a risk of hypoglycemia with weight loss. Clients with diabetes should be encouraged to speak to their provider about increasing capillary blood glucose testing and given parameters for when to call the provider. Abrupt withdrawal of the drug due to the topiramate component has been associated with seizures in clients without a history of seizures or epilepsy. In situations where immediate termination of phentermine plus topiramate is medically required, appropriate monitoring for seizure activity is recommended. Clients discontinuing the drug in a dosage of 15 mg/92 mg should be gradually tapered to reduce the possibility of precipitating a seizure.
Adverse effects of phendimetrazine that may occur include overstimulation of the body, which may result in restlessness, nervousness, irritability, insomnia, headache, anxiety, tachycardia/palpitations, and difficulty concentrating.
Phendimetrazine is contraindicated with concurrent use of stimulant drugs such as benzphetamine, diethylpropion, and phentermine. It also is contraindicated in clients with hypersensitivity to the drug and other CNS stimulants. Other contraindications include advanced arteriosclerosis, symptomatic cardiovascular disease, moderate and severe hypertension, hyperthyroidism, and glaucoma. The drug should not be used for highly agitated or nervous clients or for clients with a history of substance misuse.
Benzphetamine is contraindicated in pregnant and lactating clients. It is also contraindicated in clients with known hypersensitivity to the drug and other sympathomimetics, advanced arteriosclerosis, or angle-closure glaucoma.
Table 32.2 is a drug prototype table for anorexiants featuring phentermine. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects, and contraindications.
|
Drug Class
Anorexiant Mechanism of Action Causes CNS stimulation, specifically the release of norepinephrine from the hypothalamus, which in turn suppresses the appetite while increasing the basal metabolic rate, resulting in weight loss |
Drug Dosage
15–37.5 mg orally every morning. Lomaira only: 8 mg 3 times daily. Administer 30 minutes before meals or 1–2 hours after meals. |
|
Indications
For adults with obesity as evidenced by a BMI ≥30 For overweight adults with a BMI of 27–29 and with one or more comorbidities of hypertension, type 2 diabetes mellitus, or dyslipidemia For children age 12+ with a BMI in the 95th percentile on growth charts Therapeutic Effects Reduces overall weight Increases metabolism |
Drug Interactions
MAOIs Tricyclic antidepressants (TCAs) Guanethidine Kratom (an herbal supplement) Food Interactions No significant interactions |
|
Adverse Effects
Palpitations/tachycardia Hypertension Ischemia Restlessness Dizziness Insomnia Dry mouth Altered taste Constipation Stomach pain Decreased libido Impotence |
Contraindications
Hypersensitivity to sympathetic amines Glaucoma (increased intraocular pressure) Hypertension (moderate-severe) History of cardiovascular disease MAOIs within 14 days (risk of hypertensive crisis) Pregnancy/lactation (teratogenic/fatal fetal effects) Caution: Hypertension Seizures Diabetes mellitus Children under age 16 |
Nursing Implications
The nurse should do the following for clients who are taking anorexiants:
- Assess the client’s response to anorectic effects and dependency. It may become mentally and physically habit-forming (Mayo Clinic, 2023b).
- Monitor the client’s cardiovascular status frequently because stimulant effects may alter blood pressure, heart rate, exercise tolerance, and energy levels.
- Monitor for peripheral edema if the client’s cardiovascular system is compromised and pulmonary hypertension develops.
- Monitor weight status at least three times per week.
- Provide client teaching regarding the drug and when to call the health care provider. See below for client teaching guidelines.
Client Teaching Guidelines
The client taking an anorexiant should:
- Immediately report shortness of breath, chest pain, dizziness, fainting, or swelling of the extremities.
- Understand that within a few weeks, tolerance to the appetite-suppressant effects may develop. The client should not increase the dose. If this occurs, they should notify the prescriber.
- Monitor weight status at least three times per week, preferably at the same time of the day with the same amount of clothing using the same scale.
- Notify the provider of any sudden behavior or mood changes, such as mood swings and/or aggressive/angry outbursts.
- Take a missed dose as soon as possible; however, if it is close to the next dosage time, they should wait and just take the next dose.
- Be aware of the potential for misuse of the drug and that the drug can be habit-forming.
The client taking an anorexiant should not:
- Take stimulants at night because the stimulation effects may cause severe insomnia.
- Chew, crush, or open ER or SR capsules.
- Take double the dosage.
Link to Learning
National Institutes of Health Office of Dietary Supplements
Many clients do not consult their health care provider when choosing to use dietary supplements for weight loss. It is imperative that health care professionals become educated on the dietary supplements that clients may use to lose weight. The National Institutes of Health (NIH) has developed a comprehensive fact sheet for health care professionals on dietary supplements. The NIH also has a fact sheet for consumers on dietary supplements that health care professionals may find helpful in client education on the use of weight-loss supplements.
Safety Alert
Anorexiants
Clients with an eating disorder who are taking anorexiants have an increased risk of developing dependency. Also, stimulation effects on the body may lead to critical or fatal cardiovascular effects.


5.20: Discussion- Punctuation Poster Assignment
STEP 1 : Review this module and pick one punctuation or grammatical topic you find most interesting, relevant, or useful to your own life. This could be proper end punctuation, hyphens, dashes, parenthesis, brackets, ellipses, apostrophes, quotation marks, commas, semicolons, colons, fragments, or run-ons. Review this topic.
STEP 2 : Create a poster or infographic that addresses the punctuation or grammar concept you selected. This can be done as a powerpoint slide, or using a program of your choice, such as Canva, or Easelly. Include appropriate citations and references in your poster.
- Your poster must include at least three good examples of your chosen concept and at least one bad example. These should be original examples, created by you.
- Your poster must include at least two images or pictures .
STEP 3 : Next, share your creation with the world so that others can benefit from your work! Using either Flickr or Wikimedia Commons, upload your finished product and publish it so that other English instructors and students can use and reference your work. Although not required, we strongly recommend uploading and releasing your work with a Creative Commons license.
STEP 4 : Post the link to your finished poster to the discussion forum. Add in a paragraph about what you learned from the assignment and how you plan to improve your writing moving forward while being more aware of the concept you chose.
STEP 5 : Respond in two separate posts to two classmates (in at least 75 words). Explicitly address their examples and try to extend, complicate, or redirect their points in a substantive, knowledge-demonstrating way. Be sure to cite relevant concepts and key terms from the reading.
Rubric
| Criteria | Proficient | Developing | Not Evident | Points |
|---|---|---|---|---|
| Punctuation or Grammar Concept and Examples | Correctly identifies a punctuation or grammar concept from the module and explains it on the poster. There are three good examples and at least one bad example of the concept on the poster. | Does not correctly identify or describe the punctuation or grammar concept. Examples are off or incomplete. | Does not appropriately give examples of the punctuation concept. | __/8 |
| Visual Presentation | Poster is interesting and informative; includes at least two images. Generally easy to identify main points visually, and useful graphics highlight important points. Includes MLA citations or attributions for images. | Some use of visuals to display information, but layout is confusing or without clear connections. May not include two visuals or citations. | No attempt at presenting information in a creative manner. Project lacks visual cohesion. | __/8 |
| Discussion | Posts personal poster to the forum and comments on at least two classmate’s posters. | Posts poster; offers incomplete response to classmate. | Does not comment on another post. | __/4 |
| Total: | __/20 |
Contributors and Attributions
- Assignment: Punctuation. Provided by : Lumen Learning. License : CC BY: Attribution
- Modification of Intertextuality (errors added). Authored by : Vallerie Mott. License : CC BY: Attribution
- Image of Rockabye Hamlet. Authored by : OpheliaTheaterGroup. Located at : https://commons.wikimedia.org/wiki/File:Vintage_Rockabye_Poster_SMALL.jpg . License : CC BY-SA: Attribution-ShareAlike
- Image of Importance of Being Earnest. Authored by : CHSTheatre. Located at : https://commons.wikimedia.org/wiki/File:467375960_blb_3666-5(rev_0).jpg . License : CC BY-SA: Attribution-ShareAlike
- The Importance of Being Earnest. Authored by : Oscar Wilde. Located at : https://www.gutenberg.org/ebooks/844 . Project : Project Gutenberg. License : Public Domain: No Known Copyright


1.3: Fighting Intuition
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It is important to remember to fight one’s intuition when applying the scientific method. Certain theories, such as Quantum Mechanics, are very counter-intuitive. For example, in Quantum Mechanics, an object can be described as being in two locations at the same time. In the Theory of Special Relativity, it is possible for two people to disagree on whether two events occurred at the same time. These particular predictions from these theories have not been invalidated by any experiment.
There is no requirement in science that a theory be “pretty” or intuitive. The only requirement is that a theory describe experimental data. One should then take care in not forcing one’s preconceived notions into interpreting a theory. For example, Quantum Mechanics does not actually predict that objects can be in two locations at once, only that objects behave as if they were in two locations at once. A famous example is Schrödinger’s cat, which can be modeled as being both alive and dead at the same time. However, just because we model it that way does not mean that it really is alive and dead at the same time.


1.2.3: Multiplying Polynomials
By the end of this section, you will be able to:
- Multiply monomials
- Multiply a polynomial by a monomial
- Multiply a binomial by a binomial
- Multiply a polynomial by a polynomial
- Multiply special products
Before you get started, take this readiness quiz.
1. Distribute: \(2(x+3)\).
2. Simplify:
a. \(9^2\)
b. \((−9)^2\)
c. \(−9^2\)
3. Evaluate \(2x^2−5x+3\) for \(x=−2\).
Multiplying Expressions Using a Property for Exponents
Remember that a positive integer exponent indicates repeated multiplication of the same quantity. For example, in the expression \(a^m\), the positive integer exponent \(m\) tells us how many times we use the base \(a\) as a factor.
\[a^{m}= \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{\text{m factors}} \nonumber\]
For example,
\[(-9)^{5}= \underbrace{ (-9)\cdot (-9)\cdot (-9)\cdot (-9) \cdot (-9)}_{\text{5 factors}} \nonumber\]
Let’s review the vocabulary for expressions with exponents.
\(a^m = \underbrace{a\cdot a\cdot a \cdots a}_{m \text{ factors }}\)
This is read \(a\) to the \(m^{th}\) power .
In the expression \(a^m\) with positive integer \(m\) and \(a\not=0\), the exponent \(m\) tells us how many times we use the base \(a\) as a factor.
When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when we multiply and divide, the exponents may be different, and sometimes the bases may be different, too.
First, we will look at an example that leads to the Product Property for Positive Integer Exponents .
|
\(\quad x^{2} x^{3}\) |
|
|---|---|
| What does this mean? |
\(=\underbrace{x \cdot x}_{2 \text{ factors}\space} \cdot \underbrace{ x\cdot x\cdot x}_{3 \text{ factors}}\) \(=\underbrace{x\cdot x\cdot x\cdot x\cdot x}_{5 \text{ factors}}\) |
| \(= x^{5}\) |
The base stayed the same and we added the exponents.
If \(a\) is a real number and \(m\) and \(n\) are positive integers, then
\[a^ma^n=a^{m+n}. \nonumber\]
To multiply with like bases, add the exponents.
Simplify each expression:
a. \(y^5y^6\)
b. \(2^4\cdot 2^{3\cdot 4}\)
c. \(2a^7\cdot 3a\)
d. \(d^4d^5d^2\)
Solution
a.
| \(\quad y^5y^6\) | |
|---|---|
| Use the Product Property, \(a^ma^n=a^{m+n}\). | \(=y^{5+6}\) |
| Simplify. | \(=y^{11}\) |
b.
| \(\quad\; 2^4\cdot 2^{3\cdot 4}\) | |
|---|---|
| Use the Product Property, \(a^ma^n=a^{m+n}\). | \(=2^{4+3\cdot 4}\) |
| Simplify. |
\(=2^{4\cdot 4}\) \(=2^{16}\) |
c.
| \(\quad\; 2a^7\cdot 3a\) | |
|---|---|
| Rewrite, \(a=a^1\). | \(= 2a^7\cdot 3a^1\) |
|
Use the Commutative Property and
use the Product Property, \(a^ma^n=a^{m+n}\). |
\(=2\cdot 3a^{7+1}\) |
| Simplify. | \(=6a^8\) |
d.
| \(\quad d^4d^5d^2\) | |
|---|---|
| Add the exponents, since the bases are the same. | \(= d^{4+5+2}\) |
| Simplify. | \(=d^{11}\) |
Simplify each expression:
a. \(b^9b^8\)
b. \(4^{2\cdot 3}\cdot 4^3\)
c. \(3p^5\cdot 4p\)
d. \(x^6x^4x^8\)
- Answer
-
a. \(b^{17}\)
b. \(4^9\)
c. \(12p^6\)
d. \(x^{18}\)
Simplify each expression:
a. \(x^{12}x^4\)
b. \(10\cdot 10^5\)
c. \(2z\cdot 6z^7\)
d. \(b^5b^9b^5\)
- Answer
-
a. \(x^{16}\)
b. \(10^{6}\)
c. \(12z^8\)
d. \(b^{19}\)
Multiplying Monomials
We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.
Multiply:
a. \((3x^2)(−4x^3)\)
b. \(\left(\dfrac{5}{6}x^3y\right)(12xy^2)\)
Solution
a.
| \(\quad (3x^2)(−4x^3)\) | |
|---|---|
|
Use the Commutative Property and
use the Product Property, \(a^ma^n=a^{m+n}\). |
\(=3(-4)x^2x^3\) |
| Simplify. | \(=−12x^5\) |
b.
| \(\quad \left(\dfrac{5}{6}x^3y\right)(12xy^2)\) | |
|---|---|
|
Use the Commutative Property and
use the Product Property, \(a^ma^n=a^{m+n}\). |
\(=\dfrac{5}{6}\cdot 12x^3x\cdot yy^2\) |
| Simplify. | \(=10x^4y^3\) |
Multiply:
a. \((5y^7)(−7y^4)\)
b. \((25a^4b^3)(15ab^3)\)
- Answer
-
a. \(−35y^{11}\)
b. \(6a^5b^6\)
Multiply:
a. \((−6b^4)(−9b^5)\)
b. \((23r^5s)(12r^6s^7).\)
- Answer
-
a. \(54b^9\)
b. \(8r^{11}s^8\)
Multiplying a Polynomial by a Monomial
Multiplying a polynomial by a monomial is really just applying the Distributive Property of multiplication over addition and subtraction.
Multiply:
a. \(−2y(4y^2+3y−5)\)
b. \(3x^3y(x^2−8xy+y^2)\)
Solution
a.
| \(\quad −2y(4y^2+3y−5)\) | |
|---|---|
| Distribute. | \(= −2y\cdot 4y^2+(-2y)3y+(-2y)(−5))\) |
| Simplify. | \(=-8y^3-6y^2+10y\) |
b.
| \(\quad 3x^3y(x^2−8xy+y^2)\) | |
|---|---|
| Distribute. | \(=3x^3yx^2−3x^3y\cdot 8xy+3x^3yy^2)\) |
| Simplify. | \(=3x^5y-24x^4y^2+3x^3y^3\) |
Multiply:
a. \(-3y(5y^2+8y−7)\)
b. \(4x^2y^2(3x^2−5xy+3y^2)\)
- Answer
-
a. \(−15y^3−24y^2+21y\)
b. \(12x^4y^2−20x^3y^3+12x^2y^4\)
Multiply:
a. \(4x^2(2x^2−3x+5)\)
b. \(−6a^3b(3a^2−2ab+6b^2)\)
- Answer
-
a. \(8x^4−12x^3+20x^2\)
b. \(−18a^5b+12a^4b^2−36a^3b^3\)
Multiplying a Binomial by a Binomial
The multiplication of a binomial by a binomial can be performed by multiplying the each term of one binomial by the other binomial. This is seen by applying the Distributive Property twice.
\(\begin{align*} (a+b)(c+d) &= a(c+d) + b(c+d)\\ &=ac+ad+bc+bd\end{align*}\)
Multiply:
a. \((y+5)(y+8)\)
b. \((4y+3)(2y−5)\)
Solution
a.
| \(\quad(y+5)(y+8)\) | |
|---|---|
| Distribute \(y+8\). | \(=y(y+8)+5(y+8)\) |
| Distribute again. | \(=y^2+8y+5y+40\) |
| Combine like terms | \(=y^2+13y + 40\) |
Note that you could have distributed the \(y+5\) instead on the first step and proceeded from there.
b.
| \(\quad(4y+3)(2y−5)\) | |
|---|---|
| Distribute \(2y-5\). | \(=4y(2y-5)+3(2y-5)\) |
| Distribute again. | \(=8y^2-20y+6y-15\) |
| Combine like terms. | \(=8y^2-14y-15\) |
Multiply:
a. \((x+8)(x+9)\)
b. \((3c+4)(5c−2)\)
- Answer
-
a. \(x^2+17x+72\)
b. \(15c^2+14c−8\)
Multiply:
a. \((5x+9)(4x+3)\)
b. \((5y+2)(6y−3)\)
- Answer
-
a. \(20x^2+51x+27\)
b. \(30y^2−3y−6\)
If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the first terms in each binomial. The second and third terms are the product of multiplying the two outer terms and then the two inner terms. And the last term results from multiplying the two last terms,
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.
Let’s multiply \((x+3)(x+7)\) using the Distributive Property ( or FOIL).
| Distribute Property | FOIL |
|---|---|
| \(\quad (x+3)(x+7)\) | \(\quad (x+3)(x+7)\) |
| \(= x(x+7)+3(x+7)\) | \(=\underbrace{x^2}_{\text{F}}+\underbrace{7x}_{\text{O}}+\underbrace{3x}_{\text{I}}+\underbrace{21}_{\text{L}}\) |
| \(=x^2+7x+3x+21\) | \(=x^2+10x+21\) |
| \(=x^2+10x+21\) |
The following chart explains in details how to multiply binomials using FOIL. Note that FOIL does not work with other polynomials (why?)!
| \(\quad (a+b)(c+d)\) | |
|---|---|
| Identify first, outer, inner and last terms. |
\(= (\underbrace{\overbrace{a}^{\text{outer}}}_{\text{first}}+\underbrace{\overbrace{b}^{\text{inner}}}_{\text{last}})(\underbrace{\overbrace{c}^{\text{inner}}}_{\text{first}}+\underbrace{\overbrace{d}^{\text{outer}}}_{\text{last}})\) |
| Multiply the first terms. (F) | \(=\underbrace{ac}_{\text{F}} + \underbrace{\_\_}_{\text{O}} + \underbrace{\_\_}_{\text{I}}+ \underbrace{\_\_}_{\text{L}}\) |
| Multiply the outer terms. (O) | \(=\underbrace{ac}_{\text{F}} + \underbrace{ad}_{\text{O}} + \underbrace{\_\_}_{\text{I}} +\underbrace{\_\_}_{\text{L}}\) |
| Multiply the inner terms. (I) |
\(=\underbrace{ac}_{\text{F}} + \underbrace{ad}_{\text{O}} +\underbrace{bc}_{\text{I}} +\underbrace{\_\_}_{\text{L}}\) |
| Multiply the last terms.(L) | \(=\underbrace{ac}_{\text{F}} + \underbrace{ad}_{\text{O}} + \underbrace{bc}_{\text{I}} +\underbrace{bd}_{\text{L}}\) |
| Combine like terms, when possible. | \(=ac+ad+bc+bc\) |
When you multiply using FOIL, drawing the lines will help your brain focus on the pattern and make it easier to apply.
Now we will do an example where we use the FOIL pattern to multiply two binomials.
Multiply:
a. \((y−7)(y+4)\)
b. \((4x+3)(2x−5)\)
Solution
a.
| \(\quad (y-7)(y+4)\) | |
|---|---|
| Identify first, outer, inner and last terms. |
\(= (\underbrace{\overbrace{y}^{\text{outer}}}_{\text{first}}+\underbrace{\overbrace{(-7)}^{\text{inner}}}_{\text{last}})(\underbrace{\overbrace{y}^{\text{inner}}}_{\text{first}}+\underbrace{\overbrace{4}^{\text{outer}}}_{\text{last}})\) |
| Multiply the first terms. (F) | \(=\underbrace{y^2}_{\text{F}} + \underbrace{\_\_}_{\text{O}} + \underbrace{\_\_}_{\text{I}}+ \underbrace{\_\_}_{\text{L}}\) |
| Multiply the outer terms. (O) | \(=\underbrace{y^2}_{\text{F}} + \underbrace{4y}_{\text{O}} + \underbrace{\_\_}_{\text{I}} +\underbrace{\_\_}_{\text{L}}\) |
| Multiply the inner terms. (I) |
\(=\underbrace{y^2}_{\text{F}} + \underbrace{4y}_{\text{O}} +\underbrace{(-7y)}_{\text{I}} +\underbrace{\_\_}_{\text{L}}\) |
| Multiply the last terms.(L) | \(=\underbrace{y^2}_{\text{F}} + \underbrace{4y}_{\text{O}} + \underbrace{(-7y)}_{\text{I}} +\underbrace{(-28)}_{\text{L}}\) |
| Combine like terms, when possible. |
\(=y^2+4y-7y-28\) \(=y^2-3y-28\) |
b.
| \(\quad(4x+3)(2x−5)\) | |
|---|---|
| Identify first, outer, inner and last terms. |
\(= (\underbrace{\overbrace{4x}^{\text{outer}}}_{\text{first}}+\underbrace{\overbrace{3}^{\text{inner}}}_{\text{last}})(\underbrace{\overbrace{2x}^{\text{inner}}}_{\text{first}}+\underbrace{\overbrace{(-5)}^{\text{outer}}}_{\text{last}})\) |
| Multiply the first terms. (F) | \(=\underbrace{8x^2}_{\text{F}} + \underbrace{\_\_}_{\text{O}} + \underbrace{\_\_}_{\text{I}}+ \underbrace{\_\_}_{\text{L}}\) |
| Multiply the outer terms. (O) | \(=\underbrace{8x^2}_{\text{F}} + \underbrace{(-20x)}_{\text{O}} + \underbrace{\_\_}_{\text{I}} +\underbrace{\_\_}_{\text{L}}\) |
| Multiply the inner terms. (I) |
\(=\underbrace{8x^2}_{\text{F}} + \underbrace{(-20x)}_{\text{O}} +\underbrace{6x}_{\text{I}} +\underbrace{\_\_}_{\text{L}}\) |
| Multiply the last terms.(L) | \(=\underbrace{8x^2}_{\text{F}} + \underbrace{(-20x)}_{\text{O}} + \underbrace{6x}_{\text{I}} +\underbrace{(-15)}_{\text{L}}\) |
| Combine like terms, when possible. |
\(=8x^2+(-20x)+6x+(-15)\) \(=8x^2-14x-15\)
|
Multiply:
a. \((x−7)(x+5)\)
b. \((3x+7)(5x−2)\)
- Answer
-
a. \(x^2−2x−35\)
b. \(15x^2+29x−14\)
Multiply:
a. \((b−3)(b+6)\)
b. \((4y+5)(4y−10)\)
- Answer
-
a. \(b^2+3b−18\)
b. \(16y^2−20y−50\)
The final products in the last example were trinomials because we could combine the two middle terms. This is not always the case.
Multiply:
a. \((n^2+4)(n−1)\)
b. \((3pq+5)(6pq−11)\)
Solution
a.
| \(\quad(n^2+4)(n−1)\) | |
|---|---|
| Identify first, outer, inner and last terms. |
\(= (\underbrace{\overbrace{n^2}^{\text{outer}}}_{\text{first}}+\underbrace{\overbrace{4}^{\text{inner}}}_{\text{last}})(\underbrace{\overbrace{n}^{\text{inner}}}_{\text{first}}+\underbrace{\overbrace{(-1)}^{\text{outer}}}_{\text{last}})\) |
| Multiply the first terms. (F) | \(=\underbrace{n^3}_{\text{F}} + \underbrace{\_\_\_}_{\text{O}} + \underbrace{\_\_\_}_{\text{I}}+ \underbrace{\_\_\_}_{\text{L}}\) |
| Multiply the outer terms. (O) | \(=\underbrace{n^3}_{\text{F}} + \underbrace{(-n^2)}_{\text{O}} + \underbrace{\_\_\_}_{\text{I}} +\underbrace{\_\_\_}_{\text{L}}\) |
| Multiply the inner terms. (I) |
\(=\underbrace{n^3}_{\text{F}} + \underbrace{(-n^2)}_{\text{O}} +\underbrace{4n}_{\text{I}} +\underbrace{\_\_\_}_{\text{L}}\) |
| Multiply the last terms.(L) | \(=\underbrace{n^3}_{\text{F}} + \underbrace{(-n^2)}_{\text{O}} + \underbrace{4n}_{\text{I}} +\underbrace{(-4)}_{\text{L}}\) |
| Combine like terms, when possible. |
\(=n^3+(-n^2)+4n+(-4)\) \(=n^3-n^2+4n-4\)
|
b.
| \(\quad(3pq+5)(6pq−11)\) | |
|---|---|
| Identify first, outer, inner and last terms. |
\(= (\underbrace{\overbrace{3pq}^{\text{outer}}}_{\text{first}}+\underbrace{\overbrace{5}^{\text{inner}}}_{\text{last}})(\underbrace{\overbrace{6pq}^{\text{inner}}}_{\text{first}}+\underbrace{\overbrace{(-11)}^{\text{outer}}}_{\text{last}})\) |
| Multiply the first terms. (F) | \(=\underbrace{15p^2q^2}_{\text{F}} + \underbrace{\_\_\_}_{\text{O}} + \underbrace{\_\_\_}_{\text{I}}+ \underbrace{\_\_\_}_{\text{L}}\) |
| Multiply the outer terms. (O) | \(=\underbrace{15p^2q^2}_{\text{F}} + \underbrace{(-33pq)}_{\text{O}} + \underbrace{\_\_\_}_{\text{I}} +\underbrace{\_\_\_}_{\text{L}}\) |
| Multiply the inner terms. (I) |
\(=\underbrace{15p^2q^2}_{\text{F}} + \underbrace{(-33pq)}_{\text{O}} +\underbrace{30pq}_{\text{I}} +\underbrace{\_\_\_}_{\text{L}}\) |
| Multiply the last terms.(L) | \(=\underbrace{15p^2q^2}_{\text{F}} + \underbrace{(-33pq)}_{\text{O}} + \underbrace{30pq}_{\text{I}} +\underbrace{(-55)}_{\text{L}}\) |
| Combine like terms, when possible. |
\(=15p^2q^2+(-33pq)+30pq+(-55)\) \(=15p^2q^2-3pq+4n-55\)
|
Multiply:
a. \((x^2+6)(x−8)\)
b. \((2ab+5)(4ab−4)\)
- Answer
-
a. \(x^3−8x^2+6x−48\)
b. \(8a^2b^2+12ab−20\)
Multiply:
a. \((y^2+7)(y−9)\)
b. \((2xy+3)(4xy−5)\)
- Answer
-
a. \(y^3−9y^2+7y−63\)
b. \(8x^2y^2+2xy−15\)
The FOIL method is just one way to keep track of the four terms that appear in the product of two binomials, and it only works for binomials. Another way for obtaining the four term is using the Vertical format. It is very much like the procedure you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
\[\begin{align*}&32\\ \times &35\\ \hline\\ &160\\+&96*\\ \hline \\ &1120\end{align*}\]
Now we’ll apply this same procedure to multiply two binomials.
Multiply \((3y−1)(2y−6)\) using the Vertical format.
Solution
It does not matter which binomial goes on the top.
| \(\quad(3y−1)(2y−6)\) | |
|
Multiply \(3y−1\) by \(−6\). Multiply \(3y−1\) by \(2y\). |
\(-18y+6\) \(+6y^2-2y\) |
| Add like terms. | \(=6y^2-20y+6\) |
Multiply \((5m−7)(3m−6)\) using the Vertical format.
- Answer
-
\(15m^2−51m+42\)
Multiply \((6b−5)(7b−3)\) using the Vertical format.
- Answer
-
\(42b^2−53b+15\)
Multiplying a Polynomial by a Polynomial
We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we use the Distributive Property. There are various ways to keep track of the terms. The first is to just note one way to see the why we can distribute!
Consider finding the area of the marked figure below:
One way to calculate the area is to find the width (\(a+b\)) and the length (\(A+B+C\)) and then multiply to get:
\((a+b)(A+B+C)\)
The other way is to find the area of each little rectangle (indicated here inside the corresponding rectangle)
and then add the results (in whichever order we like) to get, for example, that the area is also equal to
\(Aa+Ab+Ba+Bb+Ca+Cb\)
Notice that we have in this example, six terms, each of which is the result of the product of a length from the first grouping times a length of the second grouping.
Since the area is the same no matter how we compute it, we have, for example,
\((a+b)(A+B+C)=Aa+Ab+Ba+Bb+Ca+Cb.\)
The picture only is valid for positive values since only such values represent length, but the equation is true for all values (and a picture can be drawn to specifically indicate negative quantities as length subtraction). The picture here then can generally be used for bookkeeping purposes no matter if the values of \(a, b, A, B,\) and \(C\) are all positive.
Multiply \((b+3)(2b^2−5b+8)\).
Solution
We present three solutions.
First, we create the rectangle and labeling above in this case for the purpose of book-keeping (and ignore the fact that negative values can not be lengths):
To find \(b+3)(2b^2−5b+8)=2b^3+6b^2-5b^2-15b+8b+24=2b^3+b^2-7b+24\)
(the last equality we see by collecting like terms).
The second way is by using the Distributive Property of multiplication of monomials over addition/subtraction.
| \(\quad (b+3)(2b^2−5b+8)\) | |
|---|---|
| Distribute. | \(=b(2b^2−5b+8) +3(2b^2−5b+8)\) |
| Distribute. |
\(=2b^3-5b^2+8b+6b^2-15b+24\) |
| Combine like terms. |
\(=2b^3+b^2-7b+24\) |
The third way is using the vertical format. It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.
| \(\quad (b+3)(2b^2−5b+8)\) | ||
|---|---|---|
|
Multiply \((2b^2−5b+8)\) by 3.
Multiply \((2b^2−5b+8)\) by \(b\). |
\(=6b^2-15b+24\) \(+2b^3-5b^2+8b\) |
|
| Add like terms. | \(=2b^3+b^2-7b+24\) |
Multiply \((y−3)(y^2−5y+2)\).
- Answer
-
a. \(y^3−8y2+17y−6\)
b. \(y^3−8y2+17y−6\)
Multiply \((x+4)(2x^2−3x+5)\).
- Answer
-
a. \(2x^3+5x^2−7x+20\)
b. \(y^3−8y^2+17y−6\)
Multiplying Special Products
Sometimes identifying patterns will make computations 'easier'. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying them, there is less work to do if you learn to use a pattern. Let’s start by looking at three examples and look for a pattern.
Look at these results. Do you see any patterns?
\[(x-5)^2=x^2-10x+25, \quad (2y+3)^2=4y^2+12y+9, \quad (5z-2t)^2=25z^2-20zt+4t^2\nonumbe\]
What about the number of terms? In each example we squared a binomial and the result was a trinomial.
\[(a+b)^2=\text{___}+\text{___}+\text{___} \nonumber\]
Now look at the first term in each result. Where did it come from?
The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!
\[(a+b)^2=a^2+\text{___}+\text{___} \nonumber\]
To get the first term of the product, square the first term.
Where did the last term come from? Look at the examples and find the pattern.
The last term is the product of the last terms, which is the square of the last term.
\[(a+b)^2=\text{___}+\text{___}+b^2 \nonumber\]
To get the last term of the product, square the last term.
Finally, look at the middle term . Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.
\[(a+b)^2=\text{___}+2ab+\text{___} \nonumber\]
\[(a−b)^2=\text{___}−2ab+\text{___} \nonumber\]
To get the middle term of the product, multiply the terms and double their product.
Putting it all together:
If \(a\) and \(b\) are real numbers,
|
\(\underbrace{(a+b)^2}_{\text{(binomial)}^2}=\underbrace{a^2}_{(\text{first term})^2}+\underbrace{2ab}_{2(\text{product of terms})}+\underbrace{b^2}_{\text{(last term)}^2},\) \(\underbrace{(a-b)^2}_{\text{(binomial)}^2}=\underbrace{a^2}_{(\text{first term})^2}-\underbrace{2ab}_{2(\text{product of terms})}+\underbrace{b^2}_{\text{(last term)}^2}.\) |
|---|
To square a binomial, square the first term, square the last term, double their product.
Multiply:
a. \((x+5)^2\)
b. \((2x−3y)^2\)
Solution
a.
| \(\quad(x+5)^2\) | |
|---|---|
| Identify \(a\) and \(b\). | \(=(\underbrace{x}_{a}+\underbrace{5}_{b})^2\) \(\quad\quad a=x,\;b=5\) |
| Square the first term, \(a\). | \(=\underbrace{x^2}_{a^2}+\underbrace{\_\_\_}_{2ab}+\underbrace{\_\_\_}_{b^2}\) |
| Double the product of \(a\) and \(b\). | \(=\underbrace{x^2}_{a^2}+\underbrace{2\cdot (x)\cdot (5)}_{2ab}+\underbrace{\_\_\_}_{b^2}\) |
| Square the last term, \(b\). | \(=\underbrace{x^2}_{a^2}+\underbrace{2\cdot (x)\cdot (5)}_{2ab}+\underbrace{5^2}_{b^2}\) |
| Simplify. | \(=x^2+10x+25\) |
b.
| \(\quad(2x-3y)^2\) | |
|---|---|
| Identify \(a\) and \(b\). | \(=(\underbrace{2x}_{a}-\underbrace{3y}_{b})^2\)\(\quad\quad a=2x,\; b=3y\) |
| Square the first term, \(a\). | \(=\underbrace{(2x)^2}_{a^2}-\underbrace{\_\_\_}_{2ab}+\underbrace{\_\_\_}_{b^2}\) |
| Double the product of \(a\) and \(b\). | \(=\underbrace{(2x)^2}_{a^2}-\underbrace{2\cdot (3x)\cdot (3y)}_{2ab}+\underbrace{\_\_\_}_{b^2}\) |
| Square the last term, \(b\). | \(=\underbrace{x^2}_{a^2}-\underbrace{2\cdot (3x)\cdot (3y)}_{2ab}+\underbrace{(3y)^2}_{b^2}\) |
| Simplify. | \(=4x^2-12xy+9y^2\) |
Multiply:
a. \((x+9)^2\)
b. \((2c−d)^2\)
- Answer
-
a. \(x^2+18x+81\)
b. \(4c^2−4cd+d^2\)
Multiply:
a. \((y+11)^2\)
b. \((4x−5y)^2\)
- Answer
-
a. \(y^2+22y+121\)
b. \(16x^2−40xy+25y^2\)
We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.
A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form \((a−b)\), \((a+b)\).
A conjugate pair is two binomials of the form
\[(a−b), (a+b). \nonumber\]
The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.
There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier. Let’s look for the pattern by using FOIL to multiply some conjugate pairs.
\[(x-4)(x+4)=x^2+4x-4x-16=x^2-16\nonumber\]
\[(2x+3)(2x-3)=4x^2-6x+6x-9=4x^2-9\nonumber\]
\[(3z-2t)(3z+2t)=9z^2+6zt-6tz-4t^2=9z^2-4t^2\nonumber\]
What do you observe about the products?
The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.
Each first term is the product of the first terms of the binomials, and since they are identical it is the square of the first term.
\[(a+b)(a−b)=a^2−\text{___} \nonumber\]
To get the first term, square the first term.
The last term came from multiplying the last terms, the square of the last term.
\[(a+b)(a−b)=a^2−b^2 \nonumber\]
To get the last term, square the last term .
Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.
The product of conjugates is always of the form \(a^2−b^2\). This is called a difference of squares .
This leads to the pattern:
For any \(a\) and \(b\),
\[(a+b)(a-b)=a^2-b^2\nonumber\]The product is called a difference of squares .
To multiply conjugates, square the first term, square the last term, write it as a difference of squares.
Multiply using the product of conjugates pattern:
a. \((2x+5)(2x−5)\)
b. \((5m−9n)(5m+9n)\)
Solution
a.
| \(\quad (2x+5)(2x−5)\) | |
|---|---|
|
Are the binomials conjugates? Identify \(a\) and \(b\) |
The binomials \(2x+5\) and \(2x-5\) are conjugates. \(\quad\quad a=2x,\;b=5\) |
| It is the product of conjugates. | \(=(\underbrace{2x+5}_{a+b})(\underbrace{2x-5}_{a-b})\) |
| Square the first term, \(a\). | \(=\underbrace{(2x)^2}_{a^2}-\underbrace{\_\_\_}_{b^2}\) |
| Square the last term, \(b\). | \(=\underbrace{(2x)^2}_{a^2}-\underbrace{(5)^2}_{b^2}\) |
|
Simplify. The product is a difference of squares. |
\(=4x^2-25\) |
b.
| \(\quad (5m-9n)(5m+9n)\) | |
|---|---|
|
Are the binomials conjugates? Identify \(a\) and \(b\). |
The binomials \(5m-9n\) and \(5m+9n\) are conjugates. \(\quad\quad a = 5m, \; b=9n\) |
| It is the product of conjugates. | \(=(\underbrace{5m-9n}_{a-b})(\underbrace{5m+9n}_{a+b})\) |
| Square the first term, \(a\). | \(=\underbrace{(5m)^2}_{a^2}-\underbrace{\_\_\_}_{b^2}\) |
| Square the last term, \(b\). | \(=\underbrace{(5m)^2}_{a^2}-\underbrace{(9n)^2}_{b^2}\) |
|
Simplify. The product is a difference of squares. |
\(=25m^2-81n^2\) |
Multiply:
a. \((6x+5)(6x−5)\)
b. \((4p−7q)(4p+7q)\)
- Answer
-
a. \(36x^2−25\)
b. \(16p^2−49q^2\)
Multiply:
a. \((2x+7)(2x−7)\)
b. \((3x−y)(3x+y)\).
- Answer
-
a. \(4x^2−49\)
b. \(9x^2−y^2\)
We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.
| Binomial Squares | Product of Conjugates |
|---|---|
| \((a+b)^2=a^2+2ab+b^2\) | \((a−b)(a+b)=a^2−b^2\) |
| \((a−b)^2=a^2−2ab+b^2\) | |
| • Squaring a binomial | • Multiplying conjugates |
| • Product is a trinomial | • Product is a binomial. |
| • Inner and outer terms with FOIL are the same. | • Inner and outer terms with FOIL are opposites. |
| • Middle term is double the product of the terms | • There is no middle term. |
Choose the appropriate pattern and use it to find the product:
a. \((2x−3)(2x+3)\)
b. \((5x−8)^2\)
c. \((6m+7)^2\)
d. \((5x−6)(6x+5)\)
Solution
a. \((2x−3)(2x+3)\)
These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.
| \(\quad (\underbrace{2x-3}_{a-b})(\underbrace{2x+3}_{a+b})\quad\quad a=2x,\;b=3\) | |
| Use the pattern: \((a-b)(a+b)=a^2-b^2\). | \(=\underbrace{(2x)^2}_{a^2}- \underbrace{(3)^2}_{b^2}\) |
| Simplify. | \(=4x^2-9\) |
b. \((8x−5)^2\)
We are asked to square a binomial. It fits the binomial squares pattern.
| \(\quad (\underbrace{8x-5}_{a-b})^2\quad\quad a=8x,\;b=5\) | |
| Use the pattern: \((a-b)^2=a^2-2ab+b^2\). | \(=\underbrace{(8x)^2}_{a^2} -\underbrace{2\cdot (8x)\cdot (5)}_{2ab} + \underbrace{(5)^2}_{b^2}\) |
| Simplify. | \(=64x^2+80x+25\) |
c. \((6m+7)^2\)
Again, we will square a binomial so we use the binomial squares pattern.
| \(\quad (\underbrace{6m+7}_{a+b})^2\quad\quad a=6m,\;b=7\) | |
| Use the pattern: \((a+b)^2=a^2+2ab+b^2\). | \(=\underbrace{(6m)^2}_{a^2} +\underbrace{2\cdot (6m)\cdot (7)}_{2ab} + \underbrace{(7)^2}_{b^2}\) |
| Simplify. | \(=36m^2+84m+49\) |
d. \((5x−6)(6x+5)\)
This product does not fit the patterns, so we will use FOIL.
| \(\quad (5x-6)(6x+5)\) | |
| Use FOIL. | \(=30x^2+25x−36x−30\) |
| Simplify. | \(=30x^2−11x−30\) |
Choose the appropriate pattern and use it to find the product:
a. \((9b−2)(2b+9)\)
b. \((9p−4)^2\)
c. \((7y+1)^2\)
d. \((4r−3)(4r+3)\)
- Answer
-
a. FOIL; \(18b^2+77b−18\)
b. Binomial Squares; \(81p^2−72p+16\)
c. Binomial Squares; \(49y^2+14y+1\)
d. Product of Conjugates; \(16r^2−9\)
Choose the appropriate pattern and use it to find the product:
a. \((6x+7)^2\)
b. \((3x−4)(3x+4)\)
c. \((2x−5)(5x−2)\)
d. \((6n−1)^2\)
- Answer
-
a. Binomial Squares; \(36x^2+84x+49\)
b. Product of Conjugates; \(9x^2−16\)
c. FOIL; \(10x^2−29x+10\)
d. Binomial Squares; \(36n^2−12n+1\)
- Explain how the 'FOIL' method is the same as a sequence of two distributions.
- If you multiply two binomials, is the result a binomial? Give an example.
- How many terms does a trinomial times a monomial have?
- Is it possible that the product of a binomial and a trinomial has 7 terms? Explain and give an example.
- What is the minimal amount of work you must do to determine the degree and the leading coefficient of \((5x^3+x^2-2x+1)(-2x^2-7x+2)\)? Explain and give another example of this process.
- Show using a picture that represents a possible situation that \((5-2)(6+2-3)=30+10-15-12-4+6\).
- Which method do you prefer to use when multiplying two binomials: the rectangle, the distributive property for monomials (twice), or the FOIL method? Why? Which method do you prefer to use when multiplying a polynomial by a polynomial: the rectangle, the distributive property for monomials (twice), or the Vertical Method? Why?
-
Multiply the following:
\((x+2)(x−2)\)
\((y+7)(y−7)\)
\((w+5)(w−5)\)
Explain the pattern that you see in your answers. Does that pattern persist for \((y+7)(y+7)\)? Give an example.
- Multiply \(3x^3(2x-3)(3x+1)\).
- Multiply \(3x(3x^3y)(2x^3y^4)\).
Key Concepts
-
How to use the FOIL method to multiply two binomials.
\[(a+b)(c+d)=ac+ad+bc+bd\] -
Multiplying Two Binomials:
To multiply binomials, use the:
- Distributive Property
- FOIL Method
-
Multiplying a Polynomial by a Polynomial:
To multiply a trinomial by a binomial, use the:
- Distributive Property
- Vertical setup
-
Binomial Squares Pattern
If a and b are real numbers, \((a+b)^2=a^2+2ab+b^2\) -
Product of Conjugates Pattern
If a, b are real numbers
\[(a+b)(a-b)=a^2-b^2\]
The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, write it as a difference of squares. -
Comparing the Special Product Patterns
Binomial Squares Product of Conjugates \((a+b)^2=a^2+2ab+b^2\) \((a−b)^2=a^2−2ab+b^2\) \((a−b)(a+b)=a^2−b^2\) • Squaring a binomial • Multiplying conjugates • Product is a trinomial • Product is a binomial. • Inner and outer terms with FOIL are the same. • Inner and outer terms with FOIL are opposites. • Middle term is double the product of the terms • There is no middle term.


3.9: The Mauryan Empire (321-184 BCE)
The kingdom of Magadha was the most powerful state in India when the Nanda Dynasty came to power in 364 BCE. Nine Nanda kings made it even greater, by improving methods of tax collection and administration, funding irrigation projects and canal building, and maintaining an impressive army of infantry, cavalry, elephants, and chariots.
But Nanda aspirations were cut short when they were overthrown by Chandragupta Maurya (r. 321 – 297 BCE), who began a new period in India’s history. He and his son Bindusara (r. 287 – 273 BCE) and grandson Ashoka (r. 268 – 232 BCE) were destined to forge the first large empire in India’s history, one that would inspire the imagination of later empire builders in South Asia. The Mauryan Empire included most of the subcontinent and lasted for 140 years (see Map \(\PageIndex{1}\))
Conflicting accounts make it difficult to say anything definitive about the first two kings. Chandragupta may have come from a Kshatriya (warrior) clan, or a Vaishya (commoner) clan of peacock-tamers. In his youth, he spent time in the northwest, where he encountered Alexander the Great. With the assistance of Kautilya, a disloyal Brahmin of the Nanda court, Chandragupta formed alliances with Nanda enemies, overthrowing them in 321 BCE. Thereafter, through diplomacy and war, he secured control over central and northern India.
Kautilya, whose advice may have been critical to Chandragupta’s success, is viewed as the author of the Arthashastra , a treatise on statecraft. This handbook for kings covered in detail the arts of governing, diplomacy, and warfare. To help ensure centralization of power in the ruler’s hands, it provided a blueprint of rules and regulations necessary to maintain an efficient bureaucracy, a detailed penal code, and advice on how to deploy spies and informants.
Chandragupta’s campaigns ended when he concluded a treaty with Seleucus Nicator in 301 BCE. After Alexander the Great retreated from India and then died, a struggle for his empire broke out among his generals. Seleucus was one of them. He gained control of the eastern half and sought to reclaim northwest India. But he was confronted by Chandragupta, defeated, and forced to surrender the Indus Basin and much of Afghanistan, giving the Mauryan Empire control over trade routes to West Asia. The treaty, however, established friendly relations between the two rulers, for in exchange for hundreds of elephants, Seleucus gave Chandragupta a daughter in marriage and dispatched an envoy to his court. Hellenistic kings (see Chapter 5) maintained commercial and diplomatic ties with India.
Military expansion continued under Bindusara and Ashoka until all but the tip of the subcontinent came under the empire’s control. With King Ashoka, however, warfare came to an end. We know far more about him because he left behind a fascinating record telling much about his ideas on governing. He had edicts inscribed on rocks throughout the realm and on sandstone pillars erected in the Ganges heartland (see Map \(\PageIndex{1}\)). He then placed them in populous areas where people usually gathered, so that his officials could read them to his largely illiterate subjects.
One rock edict speaks to why King Ashoka decided to renounce violence. While waging war against a small state located along the east coast, he was deeply disturbed by the amount of suffering and dislocation the war heaped upon innocent people’s lives. This realization caused him to redouble his faith in the Buddha. Ashoka, it turns out, was a lay follower of Buddhism.
In his edicts, he proclaimed to his subjects that the sound of the drum would be replaced by the sound of the dharma . In ancient India, dharma meant universal law. For the Brahmin priests, for example, dharma meant a society and religious order founded on Vedic principles and the caste system. For Buddhists, dharma consisted of the truths taught by the Buddha. For kings, dharma was enlightened governing and just rule. Thus, Ashoka was proclaiming that he would now rule by virtue, not force.
Ashoka’s kingly dharma was shaped by his personal practice of Buddhism. This dharma consisted of laws of ethical behavior and right conduct fashioned from Indian traditions of kingship and his understanding of Buddhist principles. To gain his subject’s willing obedience, he sought to inspire a sense of gratitude by presenting himself in the role of a father looking out for his children. He told his subjects that he was appointing officers to tour his realm and attend to the welfare and happiness of all. Justice was to be impartially administered and medical treatment provided for animals and humans. A principle of non-injury to all beings was to be observed. Following this principle meant not only renouncing state violence, but also forbidding slaughtering certain animals for sacrifices or for cooking in the royal kitchen. Ashoka also proclaimed that he would replace his pleasure and hunting tours with dharma tours. During these, he promised to give gifts to Brahmins and the aged and to visit people in the countryside.
In return, Ashoka asked his subjects to observe certain principles. He knew his empire was pluralistic, consisting of many peoples with different cultures and beliefs. He believed that if he instilled certain values in these peoples, then his realm might be knit together in peace and harmony. Thus, in addition to non-injury, Ashoka taught forbearance. He exhorted his subjects to respect parents, show courtesy to servants, and, more generally, be liberal, compassionate, and truthful in their treatment of others. These values were also to be embraced by religious communities, since Ashoka did not want people fighting over matters of faith.
The king’s writ shaped the government because kings were the heart of it. They were advised by a council of ministers and served by high officials who oversaw the major functions of the state. The Mauryans were particularly concerned with efficient revenue collection and uniform administration of justice. To that end, they divided the empire into a hierarchy of provinces and districts and appointed officials to manage matters at each level. But given such an immense empire spread over a geographically and ethnically diverse territory, the level of Mauryan control varied. Historians recognize three broad zones. The first was the metropolitan region—with its capital Pataliputra— located on the Ganges Plain. This heartland was tightly governed. The second zone consisted of conquered regions of strategic and economic importance. These provinces were placed under the control of members of the royal family and senior officials, but state formation was slower. That is, the tentacles of bureaucracy did not reach as deeply into local communities. Lastly, the third zone consisted of hinterlands sparsely populated by tribes of foragers and nomads. Here, state control was minimal, amounting to little more than establishing workable relations with chieftains.
After King Ashoka’s reign, the Mauryan Empire declined. The precise reasons for this decline are unknown. Kings enjoyed only brief reigns during the final fifty years of the empire’s existence, so they may have been weak. Since loyalty to the ruler was one element of the glue that held the centralized bureaucracy together, weak kings may explain why the political leaders of provinces pulled away and the empire fragmented into smaller states. Furthermore, the Mauryan court’s demand for revenue sufficient to sustain the government and a large standing army may have contributed to discontent. In 184 BCE, the last king was assassinated by his own Brahmin military commander, and India’s first major imperial power came to an end.


7.6: Reading- APEC
APEC is a forum for 21 Pacific Rim countries that seeks to promote free trade and economic cooperation throughout the Asia-Pacific region.
KEY TAKEAWAY
- Asia-Pacific Economic Cooperation (APEC) is a forum for 21 Pacific Rim countries that seeks to promote free trade and economic cooperation.
- APEC was established in 1989 in response to the growing interdependence of Asia-Pacific economies and the advent of regional economic blocs.
- APEC member countries include Australia, Brunei, Canada, Chile, China, Hong Kong (Hong Kong, China), Indonesia, Japan, South Korea, Mexico, Malaysia, New Zealand, Papua New Guinea, Peru, Philippines, Russia, Singapore, Taiwan (Chinese Taipei), Thailand, United States, and Vietnam.
- During the meeting in 1994 in Bogor, Indonesia, APEC leaders adopted the Bogor Goals which aim for free and open trade and investment in the Asia-Pacific by 2010, for industrialized economies and by 2020, for developing economies.
Term
- Bloc: A group of countries acting together for political or economic goals, an alliance (e.g., the eastern bloc, the western bloc, a trading bloc).
The Asia-Pacific Economic Cooperation (APEC) is a forum for 21 Pacific Rim countries (formally Member Economies) that seeks to promote free trade and economic cooperation throughout the Asia-Pacific region. Established in 1989 in response to the growing interdependence of Asia-Pacific economies and the advent of regional economic blocs (such as the European Union) in other parts of the world, APEC works to raise living standards and education levels through sustainable economic growth and to foster a sense of community and an appreciation of shared interests among Asia-Pacific countries.
Member countries are: Australia, Brunei, Canada, Chile, China, Hong Kong (Hong Kong, China), Indonesia, Japan, South Korea, Mexico, Malaysia, New Zealand, Papua New Guinea, Peru, Philippines, Russia, Singapore, Taiwan (Chinese Taipei), Thailand, United States, and Vietnam.
During the meeting in 1994 in Bogor, Indonesia, APEC leaders adopted the Bogor Goals that aim for free and open trade and investment in the Asia-Pacific by 2010, for industrialized economies and by 2020, for developing economies. In 1995, APEC established a business advisory body named the APEC Business Advisory Council (ABAC), composed of three business executives from each membereconomy. To meet the Bogor Goals, APEC carries out work in three main areas:
- Trade and investment liberalization
- Business facilitation
- Economic and technical cooperation
APEC is considering the prospects and options for a Free Trade Area of the Asia-Pacific (FTAAP), which would include all APEC member economies. Since 2006, the APEC Business Advisory Council, promoting the theory that a free trade area has the best chance of converging the member nations and ensuring stable economic growth under free trade, has lobbied for the creation of a high-level task force to study and develop a plan for a free trade area. The proposal for a FTAAP arose due to the lack of progress in the Doha round of World Trade Organization negotiations, and as a way to overcome the “spaghetti bowl” effect created by overlapping and conflicting elements of the umpteen free trade agreements. There are approximately 60 free trade agreements, with an additional 117 in the process of negotiation in Southeast Asia and the Asia-Pacific region.
GLOSSARY
Community
A group of interdependent organisms inhabiting the same region and interacting with each other. A group sharing a common understanding and often the same language, manners, tradition and law. (See civilization ).
Developing
Of a country: becoming economically more mature or advanced; becoming industrialized.
Economy
Collective focus of the study of money, currency and trade, and the efficient use of resources. The system of production and distribution and consumption. The overall measure of a currency system; as the national economy.
Free trade
Free trade is a policy by which a government does not discriminate against imports or interfere with exports by applying tariffs (to imports) or subsidies (to exports) or quotas. International trade free from government interference, especially trade free from tariffs or duties on imports.
Goal
A desired result that one works to achieve. A result that one is attempting to achieve. a result that one is attempting to achieve.
Interdependence
The condition of being interdependent.
Interest
The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed. A great attention and concern from someone or something; intellectual curiosity.
Investment
A placement of capital in expectation of deriving income or profit from its use. The expenditure of capital in expectation of deriving income or profit from its use.
Process
A series of events to produce a result, especially as contrasted to product. In reference to capabilities, a process is how the capability is executed.
Prospect
The potential things that may come to pass, often favorable. A potential customer.
Standard
Something used as a measure for comparative evaluations. A level of quality or attainment.
Task force
A group of people working toward a particular task, project, or activity, especially assigned in a particular capacity.
- International Business. Authored by : Boundless. Provided by : Boundless. Located at : https://www.boundless.com/business/ . License : CC BY-SA: Attribution-ShareAlike
- Image of APEC Members. Authored by : Cflm001. Located at : http://commons.wikimedia.org/wiki/File:APEC_members_180E.svg . License : Public Domain: No Known Copyright


1.3: What Do Geologists Do?
Geologists are involved in a range of widely varying occupations with one thing in common: the privilege and responsibility of studying this fascinating planet. In Canada, many geologists work in the resource industries, including mineral exploration and mining and energy exploration and extraction. Other major areas where geologists work include hazard assessment and mitigation (e.g., assessment of risks from slope failures, earthquakes, and volcanic eruptions); water supply planning, development, and management; waste management; and assessment of geological issues in the forest industry, and on construction projects such as highways, tunnels, and bridges. Most geologists are employed in the private sector, but many work for government-funded geological organizations, such as the Geological Survey of Canada or one of the provincial geological surveys. And of course, many geologists are involved in education at the secondary and the post-secondary levels.
Some people are attracted to geology because they like to be outdoors, and it is true that many geological opportunities involve fieldwork in places that are as amazing to see as they are interesting to study. But a lot of geological work is also done in offices or laboratories. Geological work tends to be varied and challenging, and for these reasons and many others, geologists are among those who are the most satisfied with their employment.
In Canada, most working geologists are required to be registered with an association of professional geoscientists. This typically involves meeting specific post-secondary educational standards and gaining several years of relevant professional experience under the supervision of a registered geoscientist. More information can be found at Engineers and Geoscientists British Columbia .
Media Attributions
- Figure \(\PageIndex{1}\): © Steven Earle. CC BY.


14.14: Evaluating Sources
- Examine the suitability and the trustworthiness of a source
As you gather sources for your research, you’ll need to know how to assess the validity and reliability of the materials you find.
Keep in mind that the sources you find have all been put out there by groups, organizations, corporations, or individuals who have some motivation for getting this information to you. To be a good researcher, you need to learn how to assess the materials you find and determine their reliability—before deciding if you want to use them and, if so, how you want to use them.
Whether you are examining material in books, journals, magazines, newspapers, or websites, you want to consider several issues before deciding if and how to use the material you have found.
The two main questions you should ask yourself when evaluating sources are:
- Is this source suitable? (Is it relevant to my research question?)
- Is this source trustworthy? ( It is credible—should I believe it? )
Not every suitable source is trustworthy, and not every trustworthy source is suitable. In Marvin’s example that follows, you’ll see that the online professor encourages Marvin to talk to the right sources. Remember that Marvin already learned about the importance of walking to the right places to find good sources. Now, Marvin needs to talk with his sources in order to become a part of the conversation on his topic. To do this, he needs to find authors who are trustworthy and knowledgeable.
Let’s revisit Marvin and see how he can gather information about his topic by “talking” and joining the dialogue surrounding bottled water. Then, he needs to really “listen” to his sources to determine if they are trustworthy.
Marvin: If I used a university or government website on bottled water quality, readers would trust me more than if I just used a bottled water company website?
O-Prof: Yes. But to dig deeper into the question of trust, let’s move on to a second metaphor: talking . Although the metaphor of walking is useful for understanding how to find and document sources, it can give the impression that sources are separate, inert, and neutral things, waiting to be snatched up like gold nuggets and plugged into your writing. In reality, sources are parts of overlapping knowledge networks that connect meanings and the people that make and use them. Knowledge networks are always in flux, since people are always making new meaning. Let’s go back to your health and environment project. Refresh my memory. What kinds of questions do you need answers to before you can write your paper?
Marvin: Well, I need to know if bottled water is truly healthier, like the beverage companies claim. Or would I be just as well off drinking tap water?
O-Prof: To answer this question, you’ll want to find out who’s talking about these issues. As Kenneth Burke put it, you can think of sources as voices in an ongoing conversation about the world:
Imagine that you enter a parlor. You come late. When you arrive, others have long preceded you, and they are engaged in a heated discussion, a discussion too heated for them to pause and tell you exactly what it is about. In fact, the discussion had already begun long before any of them got there, so that no one present is qualified to retrace for you all the steps that had gone before. You listen for a while, until you decide that you have caught the tenor of the argument; then you put in your oar. Someone answers; you answer him; another comes to your defense; another aligns himself against you, to either the embarrassment or gratification of your opponent, depending upon the quality of your ally’s assistance. However, the discussion is interminable. The hour grows late, you must depart. And you do depart, with the discussion still vigorously in progress. (110–111)
The authors of texts aren’t speaking aloud, of course, but they’re making written statements that others can “listen” and “respond” to. Knowing which texts you can trust means understanding which authors you can trust.
Marvin: How do I figure that out?
O-Prof: It helps to know who the authors are. What they’re saying. Where, when, and to whom they’re saying it. And what their purposes are. Imagine the world as divided into many parlors like the one Kenneth Burke described. You’d want to go to the parlors where people who really know something are talking about the topics you’re interested in.
Determining Suitability
Your task as a researcher is to determine the appropriateness of the information your source contains, for your particular research project. It is a simple question, really: will this source help me answer the research questions that I am posing in my project? Will it help me learn as much as I can about my topic? Will it help me write an interesting, convincing essay for my readers?
Here are some reasons to include information:
- contains facts/opinions that you need
- contains illustrations or data you need
- contains an overview to establish the context of your paper
- was written by a well-known authority or expert
- contains a point of view that illustrates something you are trying to establish
- exemplifies something
- may have a clear explanation of something
Reasons to exclude information:
- it may not be from a scholarly journal
- it may be from a scholarly journal but be too difficult for you to understand completely
- it may be out of date
- it may not have the point of view you are researching
- it may not contain any new information.
- it may be too narrow (or too broad) in coverage
Determining Trustworthiness
To determine the trustworthiness of a source, you want to ensure that a source is current, written by an expert, accurate, and unbiased. You’ll want to consider the rhetorical context of a source, including its purpose, audience, and focus.
One excellent tool to examine both the reliability and trustworthiness of a source is the C.R.A.A.P method, which stands for:
- Currency : The timeliness of the information
- Relevance : The importance of the information for your needs
- Authority : The source of the information
- Accuracy : The reliability, truthfulness, and correctness of the information
- Purpose : The reason the information exists
Sources should always be evaluated relative to your purpose. But because there often aren’t clear-cut answers when you evaluate sources, most of the time it requires you to make inferences–educated guesses from available clues – about whether to use information from particular sources.
Contributors and Attributions
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Walk, Talk, Cook, Eat: A Guide to Using Sources. Authored by : Cynthia R. Haller. Located at : http://www.saylor.org/site/wp-content/uploads/2013/01/writing-spaces-readings-on-writing-vol-2.pdf . Project : Writing Spaces: Readings on Writing Vol. 2. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Critical Thinking As It Applies to Source Evaluation. Authored by : Denise Snee, Kristen Houlton, and Nancy Heckel . Located at : lgdata.s3-website-us-east-1.amazonaws.com/docs/679/734444/Snee_2012_Research_Analysis_and_Writing.pdf. Project : Research, Analysis, and Writing. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Evaluating Information, the CRAAP method. Provided by : The University of Rhode Island. Located at : http://uri.libguides.com/start/craap . License : CC BY-SA: Attribution-ShareAlike
- Assessing Usefulness. Provided by : Virginia Tech. Located at : http://info-skills.lib.vt.edu/evaluating_info/2.html . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Evaluating Sources. Provided by : Excelsior OWL. Located at : https://owl.excelsior.edu/research/evaluating-sources/ . License : CC BY: Attribution
- Thinking Critically About Sources . Provided by : Ohio State University Libraries. Located at : https://ohiostate.pressbooks.pub/choosingsources/chapter/thinking-about-sources/ . Project : Choosing & Using Sources: A Guide to Academic Research by. License : CC BY: Attribution
- Icon of woman at desk. Authored by : Mohamed Hassan. Provided by : Pixabay. Located at : pixabay.com/illustrations/bully-harassment-workplace-work-3233568/. License : Other . License Terms : pixabay.com/service/terms/#license


11.6: Minimalism
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- Last updated
- Save as PDF
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Name |
Native Country |
|
Anne Truitt |
United States |
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John McCracken |
United States |
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Donald Judd |
United States |
Minimalism is the use of sparse design elements to create an art piece. The movement began after World War II and was prominent in New York City during the 1960s, influenced by the Bauhaus and constructivism. It started using the reductive aspect of Modernism and a reaction against Abstract Expressionism. The trend began as geometric abstraction and then evolved to minimalism. The use of colored stripes, monochromatic colors, and hard-edged format without pictures or figures was ordinary. Critics and viewers heavily criticized the Minimalism style of art. Minimalism is based on less, bareness, and less is more. It is art stripped down to the bare essentials eliminating anything that is not needed.
Anne Truitt (1921– 2004) was an American artist and created some of her most famous works in the 1960s. Truitt was a graduate from Bryn Mawr College with a degree in psychology and was a nurse briefly until she decided art was a better career. She started with figurative sculptures but refocused on geometric forms. Wanting to display space and color in her sculptures, Truitt's first minimalist sculptures were plain, painted, large pieces made from wood and acrylic. They often resembled pillars, and a cabinet maker created the forms off of her drawings. She applied gesso to the wood columns and many, many coats of acrylic paint. In between each coat of paint, Truitt would sand down the brush marks. She alternated brushstrokes from side to side or up and down, producing a final piece of a smooth plane of color. She frequently made a recessed platform under the sculptures (14.30), so it appears to be floating above the ground.
John McCracken (1934–2011) was an American contemporary artist who developed his signature form of sculpture using a narrow plank of wood, a plank made to lean against the wall painted in a single color. McCracken thought of the plank as the base in the physical world of objects like trees and buildings, the wall representing the imagination. His sculptures were made of plywood and coated with fiberglass and resin, using color to create another dimension. The colors were bright and unusual, like bubble-gum pink. In works like the deep maroon painting Untitled (14.31), he applied the paint and sanded and polished each layer. He divided the painting into elements of shapes and symmetry, giving the composition cohesiveness.
Donald Judd (1928–1994), an American artist, critic, and sculptor, started as a painter but moved to structures with straight lines and angles. He preferred geometric forms like squares or cubes because he could use the scale and proportion of the shapes to explore space. Judd defined a new vocabulary for his forms and called them stacks, boxes, and progressions for his creations of freestanding objects. Judd liked to use metal, plywood, concrete (14.32), and Plexiglas (14.33) when he started making larger pieces for sculptured installations. He also worked with enamel on aluminum to expand his choices of color and use more than one or two colors in any individual work. Judd bought a five-story building to build large structures and prepare them for permanent installations. He was one of the most influential artists and theorists in the minimalist period.


11.2: Arc Length
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In Section 4.1 we saw that one revolution has a radian measure of \(2\pi \) rad. Note that \(2\pi\) is the ratio of the circumference (i.e. total arc length) \(C \) of a circle to its radius \(r\):
\[ \text{Radian measure of 1 revolution} ~=~ 2\pi ~=~ \frac{2\pi\,r}{r} ~=~
\frac{C}{r} ~=~ \frac{\text{total arc length}}{\text{radius}}
\nonumber \]
Clearly, that ratio is independent of \(r \). In general, the radian measure of an angle is the ratio of the arc length cut off by the corresponding central angle in a circle to the radius of the circle, independent of the radius.
To see this, recall our formal definition of a radian: the central angle in a circle of radius \(r\) which intercepts an arc of length \(r \). So suppose that we have a circle of radius \(r \) and we place a central angle with radian measure \(1 \) on top of another central angle with radian measure \(1 \), as in Figure 4.2.1(a). Clearly, the combined central angle of the two angles has radian measure \(1+1=2 \), and the combined arc length is \(r+r=2r \).
Now suppose that we cut the angle with radian measure \(1 \) in half, as in Figure 4.2.1(b). Clearly, this cuts the arc length \(r \) in half as well. Thus, we see that
\[\nonumber \begin{align}
\text{Angle} ~&=~ 1~\text{radian} \quad&\Rightarrow\quad \text{arc length} ~&=~ r ~,\\ \nonumber
\text{Angle} ~&=~ 2~\text{radians} \quad&\Rightarrow\quad \text{arc length} ~&=~ 2\,r ~,\\ \nonumber
\text{Angle} ~&=~ \tfrac{1}{2}~\text{radian} \quad&\Rightarrow\quad \text{arc length} ~&=~
\tfrac{1}{2}\,r ~,\end{align} \nonumber \]
and in general, for any \(\theta \ge 0 \),
\[\nonumber \text{Angle} ~=~ \theta~\text{radians} \quad \Rightarrow\quad \text{arc length} ~=~ \theta\,r ~,
\nonumber \]
so that
\[ \theta ~=~ \frac{\text{arc length}}{\text{radius}} ~~.
\nonumber \]
Intuitively, it is obvious that shrinking or magnifying a circle preserves the measure of a central angle even as the radius changes. The above discussion says more, namely that the ratio of the length \(s \) of an intercepted arc to the radius \(r \) is preserved, precisely because that ratio is the measure of the central angle in radians (see Figure 4.2.2).
We thus get a simple formula for the length of an arc:
In a circle of radius \(r \), let \(s \) be the length of an arc intercepted by a central angle with radian measure \(\theta \ge 0 \). Then the arc length \(s \) is:
\[ s ~=~ r\,\theta \label{4.4} \]
Example 4.3
In a circle of radius \(r=2 \) cm, what is the length \(s \) of the arc intercepted by a central angle of measure \(\theta = 1.2 \) rad?
Solution
Using Equation \ref{4.4}, we get:
\[s ~=~ r\,\theta ~=~ (2)\,(1.2) ~=~ \boxed{2.4~\text{cm}} \nonumber \]
Example 4.4
In a circle of radius \(r=10 \) ft, what is the length \(s \) of the arc intercepted by a central angle of measure \(\theta = 41^\circ\;\)?
Solution
Using Equation \ref{4.4} blindly with \(\theta = 41^\circ \), we would get \(\;s = r\,\theta = (10)\,(41) = 410 \) ft. But this impossible, since a circle of radius \(10 \) ft has a circumference of only \(2\pi\,(10) \approx 62.83 \) ft! Our error was in using the angle \(\theta \) measured in degrees , not radians . So first convert \(\theta =41^\circ \) to radians, then use \(s=r\,\theta\):
\[\theta = 41^\circ ~=~ \frac{\pi}{180} \;\cdot\; 41 ~=~ 0.716~\text{rad}
\quad\Rightarrow\quad s ~=~ r\,\theta ~=~ (10)\,(0.716) ~=~ \boxed{7.16~\text{ft}}\nonumber \]
Note that since the arc length \(s \) and radius \(r \) are usually given in the same units, radian measure is really unitless, since you can think of the units canceling in the ratio \(\frac{s}{r} \), which is just \(\theta \). This is another reason why radians are so widely used.
 What is the measure of the angle in radians? What is the measure in degrees?
Solution
Letting \(r=5 \) and \(s=2 \) in Equation \ref{4.4}, we get:
\[\theta ~=~ \frac{s}{r} ~=~ \frac{2}{5} ~=~ \boxed{0.4~\text{rad}}\nonumber \]
In degrees, the angle is:
\[ \theta = 0.4~\text{rad} ~=~ \frac{180}{\pi} \;\cdot\; 0.4 ~=~ \boxed{22.92^\circ}\nonumber \]
For central angles \(\theta > 2\pi \) rad, i.e. \(\theta > 360^\circ \), it may not be clear what is meant by the intercepted arc, since the angle is larger than one revolution and hence "wraps around'' the circle more than once. We will take the approach that such an arc consists of the full circumference plus any additional arc length determined by the angle. In other words, Equation \ref{4.4} is still valid for angles \(\theta > 2\pi \) rad.
What about negative angles? In this case using \(s=r\,\theta \) would mean that the arc length is negative, which violates the usual concept of length. So we will adopt the convention of only using nonnegative central angles when discussing arc length.
Example 4.6
A rope is fastened to a wall in two places \(8 \) ft apart at the same height. A cylindrical container with a radius of \(2 \) ft is pushed away from the wall as far as it can go while being held in by the rope, as in Figure 4.2.3 which shows the top view. If the center of the container is \(3 \) feet away from the point on the wall midway between the ends of the rope, what is the length \(L \) of the rope?
Solution:
We see that, by symmetry, the total length of the rope is \(\;L = 2\;(AB + \overparen{BC}) \). Also, notice that \(\triangle\,ADE \) is a right triangle, so the hypotenuse has length \(AE = \sqrt{DE^2 + DA^2} = \sqrt{3^2 + 4^2} = 5 \) ft, by the Pythagorean Theorem. Now since \(\overline{AB} \) is tangent to the circular container, we know that \(\angle\,ABE \) is a right angle. So by the Pythagorean Theorem we have
\[ AB ~=~ \sqrt{AE^2 - BE^2} ~=~ \sqrt{5^2 - 2^2} ~=~ \sqrt{21} ~\text{ft}. \nonumber \]
By Equation \ref{4.4} the arc \(\overparen{BC} \) has length \(BE \cdot \theta \), where \(\theta = \angle\,BEC \) is the supplement of \(\angle\,AED + \angle\,AEB \). So since
\[ \tan\,\angle\,AED ~=~ \frac{4}{3} ~\Rightarrow~ \angle\,AED ~=~ 53.1^\circ \quad\text{and}\quad
\cos\,\angle\,AEB ~=~ \frac{BE}{AE} ~=~ \frac{2}{5} ~\Rightarrow~ \angle\,AEB ~=~ 66.4^\circ ~,
\nonumber \]
we have
\[ \nonumber
\theta ~=~ \angle\,BEC ~=~ 180^\circ \;-\; (\angle\,AED + \angle\,AEB) ~=~ 180^\circ \;-\;
(53.1^\circ + 66.4^\circ) ~=~ 60.5^\circ ~.
\nonumber \]
Converting to radians, we get \(\;\theta = \frac{\pi}{180} \;\cdot\; 60.5 = 1.06 \) rad. Thus,
\[ L ~=~ 2\,(AB \;+\; \cdot \overparen{BC}) ~=~ 2\,(\sqrt{21} \;+\; BE \cdot \theta) ~=~
2\,(\sqrt{21} \;+\; (2)\,( 1.06)) ~=~ \boxed{13.4 ~\text{ft}} ~.
\nonumber \]
Example 4.7
The centers of two belt pulleys, with radii of \(5 \) cm and \(8 \) cm, respectively, are \(15 \) cm apart. Find the total length \(L \) of the belt around the pulleys.
Solution
In Figure 4.2.4 we see that, by symmetry, \(\;L = 2\;(\overparen{DE} + EF + \overparen{FG}) \).
First, at the center \(B \) of the pulley with radius \(8 \), draw a circle of radius \(3 \), which is the difference in the radii of the two pulleys. Let \(C \) be the point where this circle intersects \(\overline{BF} \). Then we know that the tangent line \(\overline{AC} \) to this smaller circle is perpendicular to the line segment \(\overline{BF} \). Thus, \(\angle\,ACB \) is a right angle, and so the length of \(\overline{AC} \) is
\[AC ~=~ \sqrt{AB^2 - BC^2} ~=~ \sqrt{15^2 - 3^2} ~=~ \sqrt{216} ~=~ 6\,\sqrt{6}\nonumber \]
by the Pythagorean Theorem. Now since \(\overline{AE} \perp \overline{EF} \) and \(\overline{EF} \perp \overline{CF} \) and \(\overline{CF} \perp \overline{AC} \), the quadrilateral \(AEFC \) must be a rectangle. In particular, \(EF = AC \), so \(EF = 6\,\sqrt{6} \).
By Equation \ref{4.4} we know that \(\;\overparen{DE} = EA \cdot \angle\,DAE\; \) and \(\;\overparen{FG} = BF \cdot \angle\,GBF \), where the angles are measured in radians. So thinking of angles in radians (using \(\pi \) rad \(= 180^\circ\)), we see from Figure 4.2.4 that
\[\nonumber \angle\,DAE ~=~ \pi \;-\; \angle\,EAC \;-\; \angle\,BAC ~=~ \pi \;-\; \frac{\pi}{2} \;-\;
\angle\,BAC ~=~ \frac{\pi}{2} \;-\; \angle\,BAC ~, \nonumber \]
where
\[\nonumber \sin\;\angle\,BAC ~=~ \frac{BC}{AB} ~=~ \frac{3}{15} ~=~ 0.2 \quad\Rightarrow\quad
\angle\,BAC ~=~ 0.201~\text{rad.} \nonumber \]
Thus, \(\;\angle\,DAE = \frac{\pi}{2} \,-\, 0.201 = 1.37 \) rad. So since \(\overline{AE} \) and \(\overline{BF} \) are parallel, we have \(\;\angle\,ABC = \angle\,DAE = 1.37 \) rad. Thus, \(\;\angle\,GBF = \pi \,-\, \angle\,ABC = \pi \,-\, 1.37 = 1.77 \) rad. Hence,
\[\nonumber L ~=~ 2\;(\overparen{DE} \;+\; EF \;+\; \overparen{FG}) ~=~ 2\;(5\;(1.37) \;+\; 6\,\sqrt{6} \;+\;
8\;(1.77)) ~=~ \boxed{71.41~\text{cm}} ~. \nonumber \]


13: Health Care Choices
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- 13.1: Understanding your Health Care Choices
- The U.S. Department of Health and Human Services (HHS) defines health literacy as "The degree to which individuals have the capacity to obtain, process, and understand basic health information and services needed to make appropriate health decisions." In a recent report by the HHS, improved consumer health literacy was determined to be an important component of health communication, medical product safety, and oral health.
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- 13.2: How to Read Health News
- If you’ve just read a health-related headline that has caused you to spit out your morning coffee, ), it’s always best to follow the Blitz slogan: “Keep Calm and Carry On”. The most important rule to remember is: don’t automatically believe the headline. It is there to draw you into buying the paper and reading the story.
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- 13.3: Health Care Choices - The Affordable Health Care Act (ACA)
- The Patient Protection and Affordable Care Act (PPACA) represents large-scale federal reform of the United States’ health care system. It expands eligibility to programs like Medicaid and CHIP, helps guarantee insurance coverage for people with pre-existing conditions, and establishes regulations to make sure that the premium funds collected by insurers and care providers go directly to medical care.
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- 13.6: Alternative Medical Practices
- Complementary and alternative medicine (CAM) is the term for medical products and practices that are not part of standard care. Standard care is what medical doctors, doctors of osteopathy and allied health professionals, such as registered nurses and physical therapists, practice. Alternative medicine means treatments that you use instead of standard ones. Complementary medicine means nonstandard treatments used along with standard ones. Examples are acupuncture, chiropractic and herbal medicin
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- 13.7: Health Disparity
- Some groups of people are affected by HIV/AIDS, viral hepatitis, STI’s, and TB, more than other groups of people. Differences may occur by gender, race or ethnicity, education, income, disability, geographic location and sexual orientation among others. Social determinants of health like poverty, unequal access to health care, lack of education, stigma, and racism are linked to health disparities.
Thumbnail: Doctor Talking with a Patient. (Public Domain; National Cancer Institute , an agency part of the National Institutes of Health ).