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Chapter 7: Problem 78

Simplify each rational expression. $$\frac{x^{3}-8}{x^{2}+2 x-8}$$

Short Answer

Expert verified
The simplified form of the rational expression \(\frac{x^{3}-8}{x^{2}+2 x-8}\) is \(\frac{x^{2}+2x+4}{x+4}\), with x 鈮� 2.

Step by step solution

01

Factor the numerator

Factoring of \(x^{3}-8\) will require the use of the difference of cubes formula, which states: \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\). Applying this formula to \(x^{3}-8\), we get \(x^{3}-8 = (x-2)(x^{2}+2x+4)\)
02

Factor the denominator

Factoring the quadratic expression \(x^{2}+2 x-8\) requires identifying two numbers that multiply to -8 and add to +2. The numbers +4 and -2 fulfill these criteria. The factored form of the denominator is then \((x-2)(x+4)\)
03

Simplification

Now substitute the factored numerator and denominator into the original expression \(\frac{x^{3}-8}{x^{2}+2 x-8} = \frac{(x-2)(x^{2}+2x+4)}{(x-2)(x+4)}\). Here, (x-2) is a common factor in both numerator and denominator. Therefore, we cancel out (x-2) from both. The simplified form of the expression is \(\frac{x^{2}+2x+4}{x+4}\) (Remember that the simplification is only valid for x 鈮� 2)

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. a. If \(y=k x\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

Use a proportion to solve each problem. According to the authors of Number Freaking, in a global village of 200 people, 9 get drunk every day. How many of the world's 6.9 billion people \((2010\) population) get drunk everyday? Round to the nearest hundredth of a billion.

What is a proportion? Give an example with your description.

After adding \(\frac{3 x+1}{4}\) and \(\frac{x+2}{4},\) I simplified the sum by dividing the numerator and the denominator by 4 I use similar procedures to find each of the following sums: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}+4 x+3}{x+2}-\frac{5 x+9}{x+2}=x-2, x \neq-2$$
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