Problem 21 - College Algebra

Question

Multiply or divide as indicated. $$\frac{x^{3}-8}{x^{2}-4} \cdot \frac{x+2}{3 x}$$

Step-by-Step Solution

Answer

The given expression simplifies to \[\frac{x}{3}+\frac{2}{3}+\frac{4}{3x}\].

Step 1: Factor the polynomials

First, factor the polynomials in the expression. The numerator \(x^{3}-8\) can be written as \((x-2)(x^{2}+2x+4)\) as these are the factors of \(x^{3}-8\). And the denominator \(x^{2}-4\) can be written as \((x-2)(x+2)\) because it is a difference of squares.

Step 2: Substitute the factored polynomials

Substitute the factored forms back into the original expression: \[\frac{(x-2)(x^{2}+2x+4)}{(x-2)(x+2)} \cdot \frac{x+2}{3 x}\]. Remember that division by a fraction is equivalent to multiplication by its reciprocal.

Step 3: Cancel out the common terms

Here, the terms \((x-2)\) and \((x+2)\) cancel each other, giving \[\frac{x^{2}+2x+4}{3x}\].

Step 4: Simplify the resulting fraction

Simplify the expression further, this will give the final answer as \[\frac{x}{3}+\frac{2}{3}+\frac{4}{3x}\].