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

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Answer 1 · 2017-03-02T16:20:54

#=(x+y)/(x-y)#

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))#

#"quadratic trinomial"/"difference of squares" div "quadratic trinomial"/"quadratic trinomial"#

#=color(blue)((2x+y)(x+2y))/color(red)((2x+y)(2x-y)) xx color(green)((x+2y)(x-y))/color(purple)((2x-y)(x+y))#

To divide by a fraction, multiply by the reciprocal:

#=color(blue)((2x+y)(x+2y))/color(red)((2x+y)(2x-y)) xx color(purple)((2x-y)(x+y))/color(green)((x+2y)(x-y))#

When you multiplying you can cancel the like factors

#=(cancel((2x+y))cancel((x+2y)))/(cancel((2x+y))cancel((2x-y))) xx (cancel((2x-y))(x+y))/(cancel((x+2y))(x-y))#

#=(x+y)/(x-y)#