How do you multiply #(10x^2-13xy-3y^2)/(8x^2-10xy-3y^2)*(2y+8x)/(2x^2+2y^2)#?

Question

Julia Adams · Accepted Answer

#{40x^3-42x^2y - 25xy^2 - 3y^3 }/{8x^4 - 10x^3y + 5x^2y^2 - 10xy^3 - 3y^4)#

#{(10x^2 - 13xy - 3y^2)(2y+8x)}/{(8x^2 - 10xy - 3y^2)(2x^2 + 2y^2)}=#

#{2y(10x^2 - 13xy - 3y^2)+8x(10x^2 - 13xy - 3y^2)}/{2x^2(8x^2 - 10xy - 3y^2) + 2y^2(8x^2 - 10xy - 3y^2))#

#{2y10x^2 - 2y13xy - 2y3y^2 +8x10x^2 - 8x13xy - 8x3y^2}/{2x^2 8x^2 - 2x^2 10xy - 2x^2 3y^2 + 2y^2 8x^2 - 2y^2 10xy - 2y^2 3y^2)#

This can be simplified to:

#{10x^2y - 13xy^2 - 3y^3 +40x^3 - 52x^2y - 12xy^2}/{8x^4 - 10x^3y - 3x^2y^2 + 8x^2y^2 - 10xy^3 - 3y^4)#

and further:

#{40x^3-42x^2y - 25xy^2 - 3y^3 }/{8x^4 - 10x^3y + 5x^2y^2 - 10xy^3 - 3y^4)#

Genesis Allen · Answer

#color(magenta)(80x^3-84x^2y-50xy^2-6y^3)/(color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4)# #(10x^2-13xy-3y^2)/(8x^2-10xy-3y^2)*(2y+8x)/(2x^2+2y^2)# #color(white)(aaaaaaaaaaaaa)##10x^2-13xy-3y^2# #color(white)(aaaaaaaaaaa)## xx underline(2y+8x)# #color(white)(aaaaaaaaaaaaa)##20x^2y-26xy^2-6y^3# #color(white)(aaaaaaaaaaa)##-104x^2y-24xy^2+0+80x^3# #color(white)(aaaaaaaaaaaaa)##overline(-84x^2y-50xy^2-6y^3+80x^3)# #color(white)(aaaaaaaaaaaaa)##color(magenta)(80x^3-84x^2y-50xy^2-6y^3# #color(white)(aaaaaaaaaaaaa)##8x^2-10xy-3y^2# #color(white)(aaaaaaaaaaa)## xx underline(2x^2+2y^2)# #color(white)(aaaaaaaaaaaaa)##16x^4-20x^3y-6x^2y^2# #color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaa)##16x^2y^2-20xy^3-6y^4# #color(white)(aaaaaaaaaaaaa)##overline(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4)# #color(white)(aaaaaaaaaaaaa)##color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4# #color(white)(aaaaaaaaaaaaa)##color(magenta)((80x^3-84x^2y-50xy^2-6y^3)/color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4#

Nathan Axel · Answer

undefined To multiply the given expression, you need to follow these steps:

1. Factorize the numerator and denominator of each fraction separately.
2. Cancel out any common factors between the numerators and denominators.
3. Multiply the remaining factors together.

Let's break down the expression and solve it step by step:

Numerator of the first fraction: 10x^2 - 13xy - 3y^2
Denominator of the first fraction: 8x^2 - 10xy - 3y^2

Numerator of the second fraction: 2y + 8x
Denominator of the second fraction: 2x^2 + 2y^2

Now, let's factorize each expression:

10x^2 - 13xy - 3y^2 can be factored as (2x - 3y)(5x + y)
8x^2 - 10xy - 3y^2 can be factored as (2x + y)(4x - 3y)

2y + 8x cannot be factored further.
2x^2 + 2y^2 can be factored as 2(x^2 + y^2)

Now, let's cancel out any common factors:

(2x - 3y)(5x + y) / (2x + y)(4x - 3y) * (2y + 8x) / 2(x^2 + y^2)

The (2x - 3y) and (2x + y) terms cancel out.

(5x + y) / (4x - 3y) * (2y + 8x) / 2(x^2 + y^2)

Now, let's multiply the remaining factors together:

(5x + y)(2y + 8x) / (4x - 3y)(2(x^2 + y^2))

Expanding the numerator and denominator:

(10xy + 40x^2 + 2y^2 + 8xy) / (8x^3 + 8xy^2 - 6x^2y - 6y^3)

Combining like terms:

(18xy + 40x^2 + 2y^2) / (8x^3 + 8xy^2 - 6x^2y - 6y^3)

Therefore, the simplified expression is (18xy + 40x^2 + 2y^2) / (8x^3 + 8xy^2 - 6x^2y - 6y^3).