When dealing with variable expressions, it's important to remember that terms with the same variable and exponent (or "like terms") can be added and subtracted like normal numbers. The terms must not only have the same variable, but also the same exponent. For example, 7x and 5x can be added to each other, but 7x and 5x2 can not.  This rule also extends to terms with multiple variables. For instance, 2xy2 can be added to -3xy2, but not -3x2y or -3y2. Let's look at the expression x2 + 3x + 6 - 8x. In this expression, we can add the 3x and -8x terms because they are like terms. Simplified, our expression is x2 - 5x + 6. Fractions that have only numbers (and no variables) in both the numerator and denominator can be simplified in several ways. First, and perhaps easiest, is to simply treat the fraction as a division problem and divide the numerator by the denominator. In addition, any multiplicative factors that appear both in the numerator and denominator can be "canceled" because they divide to give the number 1. In other words, if both the numerator and denominator share a factor, this factor can be removed from the fraction, leaving a simplified answer. For example, let's consider the fraction 36/60. If we have a calculator handy, we can divide to get an answer of .6. If we don't, however, we can still simplify by removing common factors. Another way to think of 36/60 is (6 × 6)/(6 × 10). This can be rewritten as 6/6 × 6/10. 6/6 = 1, so our expression is actually 1 × 6/10 = 6/10. However, we're not done yet - both 6 and 10 share the factor 2. Repeating the above procedure, we are left with 3/5. Variable expressions in the form of fractions offer unique opportunities for simplification. Like normal fractions, variable fractions allow you to remove factors that are shared by both the numerator and denominator. However, in variable fractions, these factors can be both numbers and actual variable expressions.  Let's consider the expression (3x2 + 3x)/(-3x2 + 15x).This fraction can be rewritten as (x + 1)(3x)/(3x)(5 - x), 3x appears both in the numerator and in the denominator. Removing these factors from the equation leaves (x + 1)/(5 - x). Similarly, in the expression (2x2 + 4x + 6)/2, since every term is divisible by 2, we can write the expression as (2(x2 + 2x + 3))/2 and thus simplify to x2 + 2x + 3. Note that you can't cancel just any term - you can only cancel multiplicative factors that appear both in the numerator and denominator. For instance, in the expression (x(x + 2))/x, the "x" cancels from both the numerator and denominator, leaving (x + 2)/1 = (x + 2). However, (x + 2)/x does not cancel to 2/1 = 2. When dealing with variable terms in parentheses with an adjacent constant, sometimes, multiplying each term in the parentheses by the constant can result in a simpler expression. This holds true for purely numeric constants and for constants that include variables.  For instance, the expression 3(x2 + 8) can be simplified to 3x2 + 24, while 3x(x2 + 8) can be simplified to 3x3 + 24x. Note that, in some cases, such as in variable fractions, the constant adjacent to the parentheses gives an opportunity for cancellation and thus shouldn't be multiplied through the parentheses. In the fraction (3(x2 + 8))/3x, for instance, the factor 3 appears both in the numerator and the denominator, so we can cancel it and simplify the expression to (x2 + 8)/x. This is simpler and easier to work with than (3x3 + 24x)/3x, which would be the answer we would get if we had multiplied through. Factoring is a technique by which some variable expressions, including polynomials, can be simplified. Think of factoring as the opposite of the "multiplying through parentheses" step above - sometimes, an expression can be rendered more simply as two terms multiplied by each other, rather than as one unified expression. This is especially true if factoring an expression allows you to cancel part of it (as you would in a fraction). In special cases (often with quadratic equations), factoring even allows you to find answers to the equation.  Let's consider the expression x2 - 5x + 6 once more. This expression can factor to (x - 3)(x - 2). So, if x2 - 5x + 6 is the numerator of a certain expression with one of these factor terms in the denominator, like is the case with the expression (x2 - 5x + 6)/(2(x - 2)), we may want to write it in factored form so that we can cancel it with the denominator. In other words, with (x - 3)(x - 2)/(2(x - 2)), the (x - 2) terms cancel, leaving us with (x - 3)/2. As hinted at above, another reason you may want to factor your expression has to do with the fact that factoring can reveal answers to certain equations, especially when those equations are written as expressions equal to 0. For instance, let's consider the equation x2 - 5x + 6 = 0. Factoring gets us (x - 3)(x - 2) = 0. Since any number times zero equals zero, we know that if we can get either of the terms of parentheses to equal zero, the whole expression on the left side of the equals sign will equal zero as well. Thus, 3 and 2 are two answers to the equation.
اجمع حدود المتغيرات المتماثلة. بسط الكسور العددية من خلال القسمة أو بطريقة "حذف" العوامل المشتركة. احذف المتغيرات المشتركة في كسور المتغيرات. اضرب الحدود التي بين الأقواس في الثوابت العددية المجاورة لها. بسط عن طريق التحليل إلى عوامل.