Problem 51 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}-8}{x-2}$$

Step-by-Step Solution

Answer

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

Step 1: Factorize the numerator

Rewrite the expression \(x^{3}-8\) as \((x)^3 - (2)^3\). A cube of two terms can be factored using the formula \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\). So, when we apply the formula to the cube difference of our expression, we obtain \((x-2)(x^{2} + 2x + 4)\).

Step 2: Cancel out common term

After the factorization of the original expression we have \(\frac{(x-2)(x^{2} + 2x + 4)}{x-2}\). We can see that \((x-2)\) in the numerator and the denominator are the same. They will cancel each other out when we divide. After that, we are left with \(x^{2} + 2x + 4\) which is already in simplified form.