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Chapter 7: Problem 25

Multiply as indicated. $$\frac{6 x+2}{x^{2}-1} \cdot \frac{1-x}{3 x^{2}+x}$$

Short Answer

Expert verified
The result of the multiplication of the two rational expressions is \[-\frac{2}{x}\]

Step by step solution

01

Simplify Each Rational Expression

In order to simplify each fraction, factor the expressions where possible. The expression \(6x+2\) can be factored to \(2(3x+1)\) and the denominator \(x^{2}-1\) can be factored with the difference of squares to \((x-1)(x+1)\). The expression \(3x^{2}+x\) can be factored to \(x(3x + 1)\). Thus the expressions are now rewritten as follows: \n \[\frac{2(3x+1)}{(x-1)(x+1)} \cdot \frac{1-x}{x(3x + 1)}\]
02

Perform the Multiplication

The multiplication of rational expressions is done by multiplying the numerators together and denominators together as follows: \n \[\frac{2(3x+1)(1-x)}{(x-1)(x+1)x(3x + 1)}\]
03

Simplify if Possible

Observe the resulting expression and look for similar terms in the numerator and denominator that can be cancelled out. Here, the terms \((3x +1)\) in the numerator and denominator can be cancelled out. The final answer after simplifying is: \n \[\frac{2(1-x)}{(x-1)x}\] Note that the factor \((1-x)\) in the numerator is similar to \((x-1)\) in the denominator, but the order is reversed, one could be simplified as the negative of the other given that \((1-x)=-1*(x-1)\). Then, after further simplification, the final answer is: \n \[-\frac{2}{x}\]

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Most popular questions from this chapter

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

When two people work together to complete a job, describe one factor that can result in more or less time than the time given by the rational equations we have been using.

Describe how to identify the corresponding sides in similar triangles.

add or subtract as indicated. Simplify the result, if possible. $$\begin{aligned} &\frac{x^{2}+9 x}{4 x^{2}-11 x-3}+\frac{3 x-5 x^{2}}{4 x^{2}-11 x-3}\\\ &x^{2}-4 x-4 x-4 \end{aligned}$$

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