Problem 50 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}+4 x^{2}-3 x-12}{x+4}$$

Step-by-Step Solution

Answer

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

Step 1: Factorizing the Cubic Numeral Polynomial

First, decompose the cubic numeral polynomial in the numerator by searching for roots. One root is easy to find: if we plug in \(x = -4\) the sum becomes zero. That tells us that \(x + 4\) is a factor of the numerator. Now, we have to divide the original cubic polynomial by \(x + 4\) to find the other factor.

Step 2: Polynomial Division

We now divide \(x^{3}+4 x^{2}-3 x-12\) by \(x+4\) using synthetic division or long division. Going through the division process, we find that the quotient is \(x^{2} - 3\).

Step 3: Simplifying the Rational Expression

Given that the numerator and denominator have common factor \(x + 4\), these cancel out, resulting in the simplified rational expression, which is \(x^{2} - 3\).