Problem 38 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x+2}{x^{2}-x-6}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression is \(\frac{1}{x-3}\).

Step 1: Factorizing the Denominator

Begin by factorizing the denominator \(x^2 -x - 6\). The factors of -6 are -1 and 6, 1 and -6, -2 and 3, or 2 and -3. The pair -3 and 2 sums to -1, which corresponds to the middle term, hence can be used to factorize. So, the factored expression will be \((x-3)(x+2)\).

Step 2: Comparing Common Factors

Examine the numerator and the denominator for common factors. Here, \(x + 2\) is a common factor in both the numerator and the denominator.

Step 3: Simplifying the expression

Cancel out the common factor of \(x + 2\) in the numerator and the denominator. This simplifies the rational expression to \(\frac{1}{x-3}\). If the numerator doesn't factorize to the same terms as the denominator or contains no common factors, then state that the rational expression cannot be simplified.