Multiplying and Dividing Rational Expressions

Examples of Rational expressions:

To Find the domain of a ration function, set the DENOMINATOR equal to zero and
solve for x . These will be the only values NOT in your domain.

Examples: Find the values that aren 't in the domain. Then write the domain in set

To get a rational expression in lowest terms, completely factor the numerator and
denominator. Then cancel any common terms .


Examples: Write each rational expression in lowest terms.

To multiply or divide rational expressions:

1. FACTOR everything completely.

2. If dividing, multiply by the reciprocal of the second term.

3. Cancel any terms that can be canceled .

4. Multiply across.


Find the reciprocal:


8.2 Adding and Subtracting Rational Expressions

To add or subtract rational expressions:

1. If the denominators are the same, add or subtract the numerators and write that
over the common denominator.

2. If the denominators are different, first find the LCD. Write each term with the
LCD as the denominator and then add or subtract.



To find the LCD:

1. Completely factor each denominator.

2. Find the product of all the different factors taking the greatest power from each

Find the LCD for the following denominators:

Adding and subtracting:

8.3 Complex Fractions

Examples of Complex Fractions:

Simplifying a Complex Fraction: Method 1.

1. Simplify the numerator and denominator separately.

2. Divide by multiplying the numerator by the reciprocal of the denominator.

3. Simplify, if possible.

Examples: Simplify.

Method 2:

1. Multiply top and bottom by the LCD of the fractions.

2. Simplify.

Examples: Simplify.

Now, pick your favorite method and simplify:

Simplify. Rid of all negative exponents.

8.4 Equations with Rational Expressions and Graphs

Before we can solve rational equations, we need to make a note of the DOMAIN of
any rational equation. This will be the intersection of the domains of each of the
individual rational expressions.

Examples: Find the domain of each equation.

To solve a rational equation:

1. Find the domain of the equation.

2. Multiply both sides by the LCD to clear fractions.

3. Now, solve for x.

4. Eliminate any solutions that are NOT in the domain.


Graphing a rational function:

1. Find the values not in the domain. For each of these values, draw a dotted vertical
line at y =that value. This is called a vertical asymptote.

2. If the degree of the denominator > degree of the numerator, then draw a dotted
line at x = 0. This is called a horizontal asymptote.

3. Plot some points using an (x; y) chart. Then, connect the dots in the "general"
rational function shape.

General rational function:


1. Graph

2. Graph

8.5 Applications of Rational Expressions


1. Suppose Find p if f = 15 and q = 25. Then find f if p = 6 and q = 9.

Remember when solving for a variable. You want that variable to be on ONLY one
side of the equation.

2. Solve for q.

3. Solve for R.

Word Problems

40. p. 462 Suppose the following triangles are similar. Find y and the lengths of the
two longest sides of each triangle.

6. p. 455 A plane travels 100 mi against the wind in the same time it takes to travel
120 mi with the wind. The wind speed is 20 mph. Find the speed of the plane in still

7. p. 456 Dona Kenly drove 300 mi north from San Antonio, mostly on the freeway.
She usually average 55 mph, but an accident slowed her speed through Dallas to 15
mph. If her trip took 6 hr, how many miles did she drive at reduced speed ?

8. p.457 a) Stan needs 45 min to do the dishes, while Deb can do them in 30 min.
How long will it take them if they work together?

b) Suppose it takes Stan 35 min to do the dishes, and together they can do them in
15 min. How long will it take Deb to do them alone?

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