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Chapter 7: Problem 69

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-1}{1-x}$$

Short Answer

Expert verified
The simplified form of the rational expression \(\frac{x^{2}-1}{1-x}\) is \(x+1\).

Step by step solution

01

Factorize the numerator

Start by factorizing the numerator. The expression \(x^{2}-1\) can be written as \((x-1)(x+1)\) as it's a difference of squares.
02

Simplify the rational expression

Now the original expression is \(\frac{(x-1)(x+1)}{1-x}\). Notice here if we flip the signs in the denominator to \(x-1\), the original expression can be written as \(\frac{(x-1)(x+1)}{x-1}\)
03

Cancel out common factors

We can cancel the common factor of \(x-1\) from the numerator and denominator. So the final simplified expression is \(x+1\)

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Most popular questions from this chapter

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-3}{y^{2}-25}+\frac{y-3}{25-y^{2}}$$

Explain how to add rational expressions when denominators are the same. Give an example with your explanation.

Factor completely: \(81 x^{4}-1\)

Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)
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