# When simplifying a rational fraction, why do you need to factor the numerator and the denominator? Include two examples!

I will be happy to critique your thinking on this.

I wouldn't ask if I knew how to guess

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## I apologize for any misunderstanding. When simplifying a rational fraction, factoring the numerator and denominator helps us find common factors that can be crossed out, which simplifies the expression further. This process is similar to canceling out common terms in algebraic equations.

Let's look at two examples to illustrate this:

Example 1:
Simplify the fraction 12/18.

Step 1: Factor the numerator and denominator:
12 = 2 Ã— 2 Ã— 3
18 = 2 Ã— 3 Ã— 3

Step 2: Cancel out common factors:
12/18 = (2 Ã— 2 Ã— 3)/(2 Ã— 3 Ã— 3) = (2/2) Ã— (2/3) Ã— (3/3) = 1 Ã— 2/3 Ã— 1 = 2/3

So, 12/18 simplifies to 2/3.

Example 2:
Simplify the fraction (x^2 - 4)/(x^2 - 9).

Step 1: Factor the numerator and denominator:
x^2 - 4 = (x - 2)(x + 2)
x^2 - 9 = (x - 3)(x + 3)

Step 2: Cancel out common factors:
(x^2 - 4)/(x^2 - 9) = (x - 2)(x + 2)/(x - 3)(x + 3)

In this example, no further simplification is possible because there are no common factors left to cancel out.

Therefore, factoring the numerator and denominator helps us simplify rational fractions by identifying and removing common factors, which results in a simplified expression.

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