Problem 31 - College Algebra

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 simplified result of the given operation is \( \frac{(x+4)(x+2)(x+6)}{x+10} \).

Step 1: Factorize the Numerators and Denominators

Begin by factorizing each of the polynomials to find their roots. The factorized form of the polynomials are: \[ \frac{(x-3)(x+4)}{(x-3)(x+10)} \cdot \frac{(x+2)(x+3)}{(x-3)(x+1)} \div \frac{x+3}{(x+1)(x+6)}\] Each term in the original expression has been replaced with a factorized equivalent.

Step 2: Multiply and Divide

Perform the multiplication and division as indicated: \[ \frac{(x-3)(x+4)(x+2)(x+3)}{(x-3)(x+10)(x-3)(x+1)} \div \frac{(x+3)}{(x+1)(x+6)} \] We can treat the division as a multiplication with the reciprocal, and then merge the multiplication: \[ \frac{(x-3)(x+4)(x+2)(x+3)(x+1)(x+6)}{(x-3)(x+10)(x-3)(x+1)(x+3)} \]

Step 3: Simplify the Expression

Now eliminate the common factors from the numerator and the denominator. The factors \(x-3\), \(x+3\) and \(x+1\) can be eliminated: \[ \frac{(x+4)(x+2)(x+6)}{x+10} \] This is the simplified result.