Problem 20 - College Algebra

Question

Multiply or divide as indicated. $$\frac{x^{2}+5 x+6}{x^{2}+x-6} \cdot \frac{x^{2}-9}{x^{2}-x-6}$$

Step-by-Step Solution

Answer

\[\frac{x^{2} - x - 6}{x - 2}\]

Step 1: Factoring Each Polynomial

To simplify the complex fractions, factor each polynomial as follows:\[\frac{x^{2}+5 x+6}{x^{2}+x-6} \cdot \frac{x^{2}-9}{x^{2}-x-6}\]can be factored into:\[\frac{(x + 2)(x + 3)}{(x - 2)(x + 3)} \cdot \frac{(x + 3)(x - 3)}{(x - 2)(x + 3)}\]

Step 2: Simplifying The Expression

After factoring, cancel out the common factors in both the numerators and denominator:\[\frac{(x + 2)(x + 3)}{(x - 2)(x + 3)} \cdot \frac{(x + 3)(x - 3)}{(x - 2)(x + 3)}\]simplifies into:\[\frac{x + 2}{x - 2} \cdot \frac{x - 3}{1}\]

Step 3: Multiplying The Fractions

Next, multiply across the numerators and the denominators:\[\frac{x + 2}{x - 2} \cdot \frac{x - 3}{1}\]gives:\[\frac{(x + 2) \cdot (x - 3)}{(x - 2) \cdot 1}\]

Step 4: Simplifying The Result

Lastly, simplify the result of multiplication as follows:\[\frac{(x + 2)(x - 3)}{(x - 2)}\]gives the final simplified form of:\[\frac{x^{2} - x - 6}{x - 2}\]