How do you multiply (2x^2+5xy+2y^2)/(4x^2-y^2)div(x^2+xy-2y^2)/(2x^2+xy-y^2)?

Mar 2, 2017

$= \frac{x + y}{x - y}$

Explanation:

The starting point with algebraic fractions is to factorise wherever possible.

color(blue)((2x^2 +5xy+2y^2))/color(red)((4x^2-y^2)) div color(green)((x^2+xy -2y^2))/(color(purple)((2x^2 +xy -y^2))

$\text{quadratic trinomial"/"difference of squares" div "quadratic trinomial"/"quadratic trinomial}$

$= \frac{\textcolor{b l u e}{\left(2 x + y\right) \left(x + 2 y\right)}}{\textcolor{red}{\left(2 x + y\right) \left(2 x - y\right)}} \times \frac{\textcolor{g r e e n}{\left(x + 2 y\right) \left(x - y\right)}}{\textcolor{p u r p \le}{\left(2 x - y\right) \left(x + y\right)}}$

To divide by a fraction, multiply by the reciprocal:

$= \frac{\textcolor{b l u e}{\left(2 x + y\right) \left(x + 2 y\right)}}{\textcolor{red}{\left(2 x + y\right) \left(2 x - y\right)}} \times \frac{\textcolor{p u r p \le}{\left(2 x - y\right) \left(x + y\right)}}{\textcolor{g r e e n}{\left(x + 2 y\right) \left(x - y\right)}}$

When you multiplying you can cancel the like factors

$= \frac{\cancel{\left(2 x + y\right)} \cancel{\left(x + 2 y\right)}}{\cancel{\left(2 x + y\right)} \cancel{\left(2 x - y\right)}} \times \frac{\cancel{\left(2 x - y\right)} \left(x + y\right)}{\cancel{\left(x + 2 y\right)} \left(x - y\right)}$

$= \frac{x + y}{x - y}$