Problem 21 - College Algebra Essentials

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 simplified form of the expression is \(\frac{x^{2}+2x+4}{3x}\).

Step 1: Factorize the polynomials

Factorize \(x^{3}-8\) into \((x-2)(x^{2}+2x+4)\)and factorize \(x^{2}-4\) into \((x-2)(x+2)\). So the problem now becomes \(\frac{(x-2)(x^{2}+2x+4)}{(x-2)(x+2)} \cdot \frac{x+2}{3x}\)

Step 2: Multiply the fractions

Now, multiply the expressions: \(\frac{(x-2)(x^{2}+2x+4)(x+2)}{(x-2)(x+2)(3x)}\)

Step 3: Simplify the expression

Simplify the above expression by canceling out common factors. The (x-2) and (x+2) factors can be removed from both the numerator and denominator.This gives: \(\frac{x^{2}+2x+4}{3x}\)