Problem 20 - College Algebra Essentials

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

The simplified result is \(x^2+6x+9\).

Step 1: Factorize the polynomials

First, four given polynomials must be factorized, if possible. \n\The factorization of the polynomial \(x^{2}+5 x+6\) is \((x+2)(x+3)\), the polynomial \(x^{2}+x-6\) is \((x-3)(x+2)\), and the polynomial \(x^{2}-9\) is a difference of squares and can be written as \((x-3)(x+3)\). The polynomial \(x^{2}-x-6\) again is \((x-3)(x+2)\).

Step 2: Substitute the factorized polynomials into the expression

Replace the original polynomials in the given expression with their factorized counterparts: So we will get \(\frac{(x+2)(x+3)}{(x-3)(x+2)} * \frac{(x-3)(x+3)}{(x-3)(x+2)}\)

Step 3: Simplify the Expression

Now we can cancel out the common factors from the numerator and denominator: The \(x+2\) and \(x-3\) will cancel themselves out and you'll be left with \((x+3)(x+3)\). Therefore, now the expression simplifies as \((x+3)^2\) or \(x^2+6x+9\).