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Chapter 7: Problem 88

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+8}{x^{3}-8}-\frac{x}{x^{3}+2 x^{2}+4 x}$$

Short Answer

Expert verified
The simplified expression is \(\frac{10x(x^2 + 2x + 4)}{(x^{3}-8)(x^{3}+2 x^{2}+4 x)}\).

Step by step solution

01

Identify the Expressions

First, let's identify the polynomial expressions. The polynomials given are \((x+8)\), \((x^{3}-8)\), \((x)\), and \((x^{3}+2 x^{2}+4 x)\).
02

Find Common Denominator

To subtract fractions, a common denominator needs to be identified. The common denominator in this case is the two exercision from each fraction i.e. \((x^{3}-8)\) and \((x^{3}+2 x^{2}+4 x)\). So the common denominator will be their product \((x^{3}-8)(x^{3}+2 x^{2}+4 x)\).
03

Subtract the Fractions

The next step is to subtract the fractions by aligning them to have the common denominator. The subtraction becomes \((x+8)(x^{3}+2 x^{2}+4 x)-(x)(x^{3}-8)\) over the common denominator.
04

Expand and Simplify

By expanding the numerators and simplifying the subtraction, the new expression is \((x^4 + 2x^3 + 4x^2 + 8x^3 + 16x^2 + 32x - x^4 + 8x)\) over the common denominator.
05

Combine Like Terms

After combining the like terms, the expression further simplifies to \((10x^3 + 20x^2 + 40x)\) over the denominator.
06

Factor Out Common Element

Lastly, factor out the common element in the numerator, which is \(10x\), to get the final simplified expression \((10x(x^2 + 2x + 4))\) over the common denominator.

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{x}+4$$

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

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Will help you prepare for the material covered in the next section. If \(B=k W,\) find the value of \(k,\) in decimal form, using \(B=5\) and \(W=160\)

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