Problem 31 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x^{2}+x-12}{x^{2}+x-30} \cdot \frac{x^{2}+5 x+6}{x^{2}-2 x-3} \div \frac{x+3}{x^{2}+7 x+6}$$

Step-by-Step Solution

Answer

The result is \(x+4 / x-5\).

Step 1: Factorize each expression

Factorize each quadratic expression involved in the exercise.\n\(\frac{{(x+4)(x-3)}}{{(x+6)(x-5)}} \cdot \frac{{(x+2)(x+3)}}{{(x+1)(x-3)}} \div \frac{{(x+3)}}{{(x+2)(x+3)}}\)

Step 2: Rewrite the division as multiplication

Rewrite the division by \(\frac{{x+3}}{{x^{2}+7x+6}}\) as the multiplication by its reciprocal, \(\frac{{x^{2}+7x+6}}{{x+3}} = \frac{{(x+2)(x+3)}}{{x+3}}\).\nThen, the expression becomes \n\(\frac{{(x+4)(x-3)}}{{(x+6)(x-5)}} \cdot \frac{{(x+2)(x+3)}}{{(x+1)(x-3)}} \cdot \frac{{(x+2)(x+3)}}{{x+3}}.\)

Step 3: Simplify

Simplify the expression by canceling out equivalent expressions that appear in the numerator and the denominator.\nAfter canceling, the expression becomes \((x+4)/(x-5)\).

Step 4: Rewriting the result

The result may be rewritten without parentheses.\nThe result is \(x+4 / x-5\).