Problem 28 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x^{2}+x}{x^{2}-4} \div \frac{x^{2}-1}{x^{2}+5 x+6}$$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{x(x+3)}{(x-2)(x-1)} \)

Step 1: Convert Division to Multiplication

Instead of dividing by the fraction, we should multiply by its reciprocal, which will simplify the problem. That changes the expression given to: \( \frac{x^{2}+x}{x^{2}-4} \times \frac{x^{2}+5x+6}{x^{2}-1} \)

Step 2: Factor Polynomials

Factor the polynomials in the numerators and denominators for possible simplification. Then the expression becomes: \( \frac{x(x+1)}{(x+2)(x-2)} \times \frac{(x+2)(x+3)}{(x+1)(x-1)} \)

Step 3: Cancel out the Common Terms

In this step, identify and cancel out any common terms in the numerator and denominator across the two fractions. The expression simplifies to: \( \frac{x}{x-2} \times \frac{x+3}{x-1} \)

Step 4: Multiply the Fractions

Multiply the numerators together and do the same for the denominators to get the final answer. The result is: \( \frac{x(x+3)}{(x-2)(x-1)} \)