Question Find the simplified product of $\frac{x+4}{x^{2}+2x-8}$ and $\frac{x-2}{3}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the product of two fractions and simplify the answer. . The fractions are $\frac{x+4}{x^{}+ x-8}$ and $\frac{x-}{3}$ .

STEP 2

First, we need to factorize the denominator of the first fraction. The quadratic expression $x^{2}+2 x-8$ can be factored into two binomial expressions.

STEP 3

Factorize the quadratic expression $x^{2}+2 x-8$ .
$x^{2}+2 x-8 = (x-2)(x+)$

STEP 4

Replace the factored expression in the first fraction.
$\frac{x+4}{x^{2}+2 x-8} = \frac{x+4}{(x-2)(x+4)}$

STEP 5

Now, we can simplify the first fraction by cancelling out the common factors in the numerator and the denominator.
$\frac{x+4}{(x-2)(x+4)} = \frac{1}{x-2}$

STEP 6

Now, we have two simplified fractions, $\frac{1}{x-2}$ and $\frac{x-2}{3}$ . We can multiply these fractions.

STEP 7

Multiply the two fractions.
$\frac{1}{x-2} \cdot \frac{x-2}{3}$

STEP 8

When multiplying fractions, we multiply the numerators together and the denominators together.
$\frac{1 \cdot (x-2)}{(x-2) \cdot3}$

STEP 9

Now, we can simplify the fraction by cancelling out the common factors in the numerator and the denominator.
$\frac{ \cdot (x-2)}{(x-2) \cdot3} = \frac{}{3}$ So, the product of the given fractions, simplified, is $\frac{}{3}$ .

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