Problem 83
Question
Simplify each rational expression. $$\frac{8 x^{2}+4 x+2}{1-8 x^{3}}$$
Step-by-Step Solution
Verified Answer
Therefore, the simplified form of the given rational expression is \( \frac{2(4x^{2}+2x+1)}{(1-2x)(1+2x+4x^{2})} \).
1Step 1: Factorizing the Numerator
To simplify the rational expression, first start with factorizing the numerator. You can factor out a 2, giving you \( 2(4x^{2}+2x+1) \).
2Step 2: 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 \( (1-2x)(1+2x+4x^2) \).
3Step 3: 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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Problem 82
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