Question

multiply as indicated.

1. $\frac{5x}{3x+6} cdot \frac{x^2 - 2x - 8}{x^2 + 2x - 8}$
2. $\frac{x^2 - 5x + 6}{x - 3} cdot \frac{x + 3}{x^2 - 9}$
3. $\frac{9y + 3}{y^2 - 9} cdot \frac{3 - y}{3y^2 + y}$

(handwritten notes below:
$3x+6=3(x+2)$
$x^2-2x-8=(x-4)(x+2)$
$x^2+2x-8=(x+4)(x-2)$
$\frac{5x}{3(x+2)} cdot \frac{(x-4)(x+2)}{(x+4)(x-2)}$

1. $\frac{5x^2 - 20x}{3x^2 + 6x - 24}$
2. $\frac{x - 2}{x - 3}$
3. ________)

Explanation:

Step1: Factor all expressions

For Problem 11:

$3x+6=3(x+2)$
$x^2-2x-8=(x-4)(x+2)$
$x^2+2x-8=(x+4)(x-2)$

For Problem 12:

$x^2-5x+6=(x-2)(x-3)$
$x^2-9=(x-3)(x+3)$

For Problem 13:

$9y+3=3(3y+1)$
$y^2-9=(y-3)(y+3)$
$3-y=-(y-3)$

Step2: Rewrite and cancel common terms

Problem 11:

$\frac{5x}{3(x+2)} \cdot \frac{(x-4)(x+2)}{(x+4)(x-2)}$
Cancel $(x+2)$:
$\frac{5x(x-4)}{3(x+4)(x-2)}$

Problem 12:

$\frac{(x-2)(x-3)}{x-3} \cdot \frac{x+3}{(x-3)(x+3)}$
Cancel $(x-3)$ and $(x+3)$:
$\frac{x-2}{x-3}$

Problem 13:

$\frac{3(3y+1)}{(y-3)(y+3)} \cdot \frac{-(y-3)}{3y^2+y}$
Factor $3y^2+y=y(3y+1)$, cancel $(y-3)$ and $(3y+1)$:
$\frac{3 \cdot (-1)}{(y+3)y} = -\frac{3}{y(y+3)}$

Answer:

1. $\boldsymbol{\frac{5x(x-4)}{3(x+4)(x-2)}}$
2. $\boldsymbol{\frac{x-2}{x-3}}$
3. $\boldsymbol{-\frac{3}{y(y+3)}}$