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

Question

Kennedy Adams · Accepted Answer

#=(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)#

Isaiah Anthony · Answer

undefined To multiply the given expressions, we need to multiply the numerators and denominators separately, and then simplify the resulting expression.

First, let's multiply the numerators: 
(2x^2 + 5xy + 2y^2) * (x^2 + xy - 2y^2)

Expanding this expression, we get:
2x^4 + 7x^3y + 4x^2y^2 + 5x^3y + 17x^2y^2 + 10xy^3 + 2x^2y^2 + 7xy^3 + 4y^4

Now, let's multiply the denominators: 
(4x^2 - y^2) * (2x^2 + xy - y^2)

Expanding this expression, we get:
8x^4 + 4x^3y - 4x^2y^2 - 2x^2y^2 - xy^3 + y^4

Finally, we divide the two resulting expressions:
(2x^4 + 7x^3y + 4x^2y^2 + 5x^3y + 17x^2y^2 + 10xy^3 + 2x^2y^2 + 7xy^3 + 4y^4) / (8x^4 + 4x^3y - 4x^2y^2 - 2x^2y^2 - xy^3 + y^4)

This is the simplified expression after multiplying and dividing the given expressions.

Eva Abramson · Answer

undefined To multiply the fractions $\frac{2x^2+5xy+2y^2}{4x^2-y^2}$ and $\frac{x^2+xy-2y^2}{2x^2+xy-y^2}$, you first factorize the numerators and denominators, cancel any common factors, and then multiply the resulting expressions. The final expression would be:

$$
\frac{(2x+y)(x+2y)}{(2x+y)(2x-y)} = \frac{x+2y}{2x-y}
$$