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Chapter 7: Problem 84

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+8}{x^{2}-9}-\frac{x+2}{x+3}+\frac{x-2}{x-3}$$

Short Answer

Expert verified
The simplified expression is \(\frac{8}{x^{2}-9}\).

Step by step solution

01

Factorization

First, the denominator in the first fraction \(x^{2}-9\) can be factored since it is a difference of squares. The factored form should be \((x-3)(x+3)\). So the first fraction can be rewritten as \(\frac{x+8}{(x-3)(x+3)}\). The entire expression now looks like this: \(\frac{x+8}{(x-3)(x+3)}-\frac{x+2}{x+3}+\frac{x-2}{x-3}\)
02

Finding a common denominator

Next, find a common denominator for all fractions. It can be observed that the common denominator should be \((x-3)(x+3)\) because each denominator can evenly divide into it. Multiply the second and the third fractions by the missing factor to get a common denominator. The expression will look like this: \(\frac{x+8}{(x-3)(x+3)}-\frac{(x+2)(x-3)}{(x-3)(x+3)}+\frac{(x+2)(x+3)}{(x-3)(x+3)}\)
03

Distribute and simplify numerators

Now distribute any terms in the numerators as needed. The expression should look like this: \(\frac{x+8-(x^{2}-x-6)+(x^{2}+x-6)}{(x-3)(x+3)}\)
04

Simplify the expression

Put the expression together and simplify the numerator. Write the numerator as one polynomial and combine like terms. The simplified expression will be \(\frac{8}{(x-3)(x+3)}\)

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Most popular questions from this chapter

add or subtract as indicated. Simplify the result, if possible. $$\frac{3 y^{2}-2}{3 y^{2}+10 y-8}-\frac{y+10}{3 y^{2}+10 y-8}-\frac{y^{2}-6 y}{3 y^{2}+10 y-8}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x-3}+\frac{2}{3-x}$$

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