8.2 Multiply and divide rational expressions

Solution

Multiply the numerators and denominators.
Look for common factors, and then remove them.
Simplify.

Solution

Multiply.
Factor the numerator and denominator completely, and then remove common factors.
Simplify.

Solution

Solution

$\begin{array}{cccc}& & & \hfill \frac{n^2-7n}{n^2+2n+1}·\frac{n+1}{2n}\hfill \\ \\ \\ \text{Factor each numerator and denominator.}\hfill & & & \hfill \frac{n\left(n-7\right)}{\left(n+1\right)\left(n+1\right)}·\frac{n+1}{2n}\hfill \\ \\ \\ \begin{array}{c}\text{Multiply the numerators and the}\hfill \\ \text{denominators.}\hfill \end{array}\hfill & & & \hfill \frac{n\left(n-7\right)\left(n+1\right)}{\left(n+1\right)\left(n+1\right)2n}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \frac{\cancel{n}\left(n-7\right)\cancel{\left(n+1\right)}}{\left(n+1\right)\cancel{\left(n+1\right)}2\cancel{n}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{n-7}{2\left(n+1\right)}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{16-4x}{2x-12}·\frac{x^2-5x-6}{x^2-16}\hfill \\ \\ \\ \text{Factor each numerator and denominator.}\hfill & & & \hfill \frac{4\left(4-x\right)}{2\left(x-6\right)}·\frac{\left(x-6\right)\left(x+1\right)}{\left(x-4\right)\left(x+4\right)}\hfill \\ \\ \\ \begin{array}{c}\text{Multiply the numerators and the}\hfill \\ \text{denominators.}\hfill \end{array}\hfill & & & \hfill \frac{4\left(4-x\right)\left(x-6\right)\left(x+1\right)}{2\left(x-6\right)\left(x-4\right)\left(x+4\right)}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \left(\mathrm{-1}\right)\frac{\cancel{2}·2\cancel{\left(4-x\right)}\cancel{\left(x-6\right)}\left(x+1\right)}{\cancel{2}\cancel{\left(x-6\right)}\cancel{\left(x-4\right)}\left(x+4\right)}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill -\frac{2\left(x+1\right)}{\left(x+4\right)}\hfill \end{array}$

Solution

Factor each numerator and denominator.
Multiply the numerators and denominators.
Remove common factors.
Simplify.