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Chapter 7: Problem 68

Perform the indicated operation or operations. $$\frac{5 x^{2}-x}{3 x+2} \div\left(\frac{6 x^{2}+x-2}{10 x^{2}+3 x-1} \cdot \frac{2 x^{2}-x-1}{2 x^{2}-x}\right)$$

Short Answer

Expert verified
\(\frac{(5 x^{2}-x) \cdot (10 x^{2}+3 x-1)}{(3 x+2) \cdot (6 x^{2}+x-2)}\)

Step by step solution

01

Simplify the Expressions

Check if there is any factorization that can be done in any of the given expressions. Note that the expression \(2 x^{2}-x\) in the denominator and numerator of the third term are the same, therefore they can cancel out. Resulting in: \(\frac{5 x^{2}-x}{3 x+2} \div \frac{6 x^{2}+x-2}{10 x^{2}+3 x-1}\)
02

Convert Division into Multiplication

Now, replace the division operation with multiplication by finding the reciprocal of the second expression. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. After taking reciprocal, the expression will be: \(\frac{5 x^{2}-x}{3 x+2} \cdot \frac{10 x^{2}+3 x-1}{6 x^{2}+x-2}\)
03

Perform Multiplication

Next step is to perform the multiplication operation between the two fractions. To multiply two fractions, simply multiply the numerators together to obtain the new numerator, and multiply the denominators together to get the new denominator. This gives: \(\frac{(5 x^{2}-x) \cdot (10 x^{2}+3 x-1)}{(3 x+2) \cdot (6 x^{2}+x-2)}\)
04

Simplify if Possible

In the last step, the expression must be checked if it can be further simplified. Since no further simplification is possible, the expression in the step above is the final answer.

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Most popular questions from this chapter

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.. $$Find the missing polynomials: $\quad-\frac{3 x-12}{2 x}=\frac{3}{2}$$

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

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 \(\frac{d}{a}=\frac{e}{b}\)

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}-2}{x^{2}+6 x-7}+\frac{19-4 x}{7-6 x-x^{2}}$$
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