Problem 88 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Expressions

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

Step 2: 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 \((x^{3}-8)(x^{3}+2 x^{2}+4 x)\).

Step 3: 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.

Step 4: 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.

Step 5: Combine Like Terms

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

Step 6: 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.