Solved on Sep 30, 2023

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

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

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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Find the simplified product of x+4x2+2x−8\frac{x+4}{x^{2}+2x-8}x2+2x−8x+4​ and x−23\frac{x-2}{3}3x−2​.

STEP 1

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

STEP 2

First, we need to factorize the denominator of the first fraction. The quadratic expression x2+2x−8x^{2}+2 x-8x2+2x−8 can be factored into two binomial expressions.

STEP 3

Factorize the quadratic expression x2+2x−8x^{2}+2 x-8x2+2x−8.x2+2x−8=(x−2)(x+)x^{2}+2 x-8 = (x-2)(x+)x2+2x−8=(x−2)(x+)

STEP 4

Replace the factored expression in the first fraction.x+4x2+2x−8=x+4(x−2)(x+4)\frac{x+4}{x^{2}+2 x-8} = \frac{x+4}{(x-2)(x+4)}x2+2x−8x+4​=(x−2)(x+4)x+4​

STEP 5

Now, we can simplify the first fraction by cancelling out the common factors in the numerator and the denominator.x+4(x−2)(x+4)=1x−2\frac{x+4}{(x-2)(x+4)} = \frac{1}{x-2}(x−2)(x+4)x+4​=x−21​

STEP 6

Now, we have two simplified fractions, 1x−2\frac{1}{x-2}x−21​ and x−23\frac{x-2}{3}3x−2​. We can multiply these fractions.

STEP 7

Multiply the two fractions.1x−2⋅x−23\frac{1}{x-2} \cdot \frac{x-2}{3}x−21​⋅3x−2​

STEP 8

When multiplying fractions, we multiply the numerators together and the denominators together.1⋅(x−2)(x−2)⋅3\frac{1 \cdot (x-2)}{(x-2) \cdot3}(x−2)⋅31⋅(x−2)​

STEP 9

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

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

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 $\frac{x-}{3}$ .

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.

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

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

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

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

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

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

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 $\frac{}{3}$ .