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Chapter 7: Problem 70

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-4}{2-x}$$

Short Answer

Expert verified
The simplified rational expression is \(-(x+2)\)

Step by step solution

01

Factor the numerator

The given rational expression is \(\frac{x^{2}-4}{2-x}\). The numerator is a difference of squares and can be factored as \((x-2)(x+2)\). So, we get \(\frac{(x-2)(x+2)}{2-x}\)
02

Rewrite the denominator

Now, rewrite the denominator \(2-x\) as \(-(x-2)\). So, our rational expression becomes \(\frac{(x-2)(x+2)}{-(x-2)}\)
03

Cancel the common factors

Now we see that \(x-2\) is a common factor in the numerator and the denominator. We can cancel these factors out. Thus our expression simplifies to \(\frac{-(x+2)}{1}\) which is \(-(x+2)\)

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x}+\frac{3}{x-5}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{10}{x+3}-\frac{2}{-x-3}$$

Describe how to identify the corresponding sides in similar triangles.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used \(\frac{a}{d}=\frac{b}{e}\) to show that corresponding sides of similar triangles are proportional, but I could also use \(\frac{a}{b}=\frac{d}{e}\) or \(\frac{d}{a}=\frac{e}{b}\)

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3 x}{(x+1)^{2}}-\left[\frac{5 x+1}{(x+1)^{2}}-\frac{3 x+2}{(x+1)^{2}}\right]$$
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