Problem 78 - Introductory Algebra for College Students

Question

Simplify each rational expression. $$\frac{x^{3}-8}{x^{2}+2 x-8}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression \(\frac{x^{3}-8}{x^{2}+2 x-8}\) is \(\frac{x^{2}+2x+4}{x+4}\), with x ≠ 2.

Step 1: Factor the numerator

Factoring of \(x^{3}-8\) will require the use of the difference of cubes formula, which states: \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\). Applying this formula to \(x^{3}-8\), we get \(x^{3}-8 = (x-2)(x^{2}+2x+4)\)

Step 2: Factor the denominator

Factoring the quadratic expression \(x^{2}+2 x-8\) requires identifying two numbers that multiply to -8 and add to +2. The numbers +4 and -2 fulfill these criteria. The factored form of the denominator is then \((x-2)(x+4)\)

Step 3: Simplification

Now substitute the factored numerator and denominator into the original expression \(\frac{x^{3}-8}{x^{2}+2 x-8} = \frac{(x-2)(x^{2}+2x+4)}{(x-2)(x+4)}\). Here, (x-2) is a common factor in both numerator and denominator. Therefore, we cancel out (x-2) from both. The simplified form of the expression is \(\frac{x^{2}+2x+4}{x+4}\) (Remember that the simplification is only valid for x ≠ 2)