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Chapter 2: Problem 82

Simplify the rational expression by using long division or synthetic division. $$\frac{x^{3}+x^{2}-64 x-64}{x+8}$$

Short Answer

Expert verified
The simplified form of the given rational expression is \(x^{2}-7x-64\).

Step by step solution

01

Set up the long division

The long division starts by aligning the divisor (\(x+8\)) to the left of the division symbol and the dividend (\(x^{3}+x^{2}-64 x-64\)) under it.
02

Perform the long division

In the first step of the long division: Divide the leading term of the dividend (\(x^{3}\)) by the leading term of the divisor (\(x\)). The result is \(x^{2}\). Write this term above the division line.\nNow, multiply the divisor (\(x+8\)) by the first term of the quotient (\(x^{2}\)) and subtract the result from the dividend. This gives us a new dividend of \(-7x^{2}-64x-64\).\nRepeat this process for the new dividend. The next term in the quotient is \(-7x\), and the new dividend after subtraction becomes \(0x-64\). The process repeats once more, yielding \(0x\), a dividend of \(-64\), and a final quotient term of \(-64\).
03

Write the final answer

After all the steps, our final quotient, which is the simplified version of the original rational expression, is \(x^{2}-7x-64\).

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Most popular questions from this chapter

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