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Chapter 7: Problem 83

Simplify each rational expression. $$\frac{8 x^{2}+4 x+2}{1-8 x^{3}}$$

Short Answer

Expert verified
Therefore, the simplified form of the given rational expression is \( \frac{2(4x^{2}+2x+1)}{(1-2x)(1+2x+4x^{2})} \).

Step by step solution

01

Factorizing the Numerator

To simplify the rational expression, first start with factorizing the numerator. You can factor out a 2, giving you \( 2(4x^{2}+2x+1) \).
02

Factorize the Denominator

Next, factorize the denominator i.e., \( 1-8 x^{3} \). Recognize that this is a difference of cubes. Therefore, it can be expressed as \( (1-2x)(1+2x+4x^2) \).
03

Simplifying the Expression

Now you have got the rational expression as \( \frac{2(4x^{2}+2x+1)}{(1-2x)(1+2x+4x^{2})} \). Here, there is nothing left to factorize or simplify, so this is the final answer.

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Most popular questions from this chapter

Anthropologists and forensic scientists classify skulls using $$ \frac{L+60 W}{L}-\frac{L-40 W}{L} $$ where \(L\) is the skull's length and \(W\) is its width. a. Express the classification as a single rational expression. b. If the value of the rational expression in part (a) is less than \(75,\) a skull is classified as long. A medium skull has a value between 75 and \(80,\) and a round skull has a value over \(80 .\) Use your rational expression from part (a) to classify a skull that is 5 inches wide and 6 inches long.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{y^{2}-16}+\frac{7-y}{16-y^{2}}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y-1}-\frac{1}{1-y}$$

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

If you know how many hours it takes for you to do a job, explain how to find the fractional part of the job you can complete in \(x\) hours.
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