SOLUTION: Please simplify: (x+y)(x^2+2xy-y)^2

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Question 251796: Please simplify:
(x+y)(x^2+2xy-y)^2
Found 2 solutions by jim_thompson5910, richwmiller:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's expand %28x%5E2%2B2xy-y%29%5E2

%28x%5E2%2B2xy-y%29%5E2 Start with the given expression.


%28x%5E2%2B2xy-y%29%28x%5E2%2B2xy-y%29 Rewrite %28x%5E2%2B2xy-y%29%5E2 as %28x%5E2%2B2xy-y%29%28x%5E2%2B2xy-y%29


x%5E2%28x%5E2%2B2xy-y%29%2B2xy%28x%5E2%2B2xy-y%29-y%28x%5E2%2B2xy-y%29 Expand. Recall that %28A%2BB%2BC%29%28D%2BE%2BF%29=A%28D%2BE%2BF%29%2BB%28D%2BE%2BF%29%2BC%28D%2BE%2BF%29


Distribute.


x%5E4%2B2x%5E3y-x%5E2y%2B2x%5E3y%2B4x%5E2y%5E2-2xy%5E2-x%5E2y-2xy%5E2%2By%5E2 Multiply.


x%5E4%2B4x%5E3y-2x%5E2y%2B4x%5E2y%5E2-4xy%5E2%2By%5E2 Now combine like terms.


So %28x%5E2%2B2xy-y%29%28x%5E2%2B2xy-y%29 expands to x%5E4%2B4x%5E3y-2x%5E2y%2B4x%5E2y%5E2-4xy%5E2%2By%5E2.


In other words, %28x%5E2%2B2xy-y%29%28x%5E2%2B2xy-y%29=x%5E4%2B4x%5E3y-2x%5E2y%2B4x%5E2y%5E2-4xy%5E2%2By%5E2.


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This means that %28x%2By%29%28x%5E2%2B2xy-y%29%5E2 then becomes %28x%2By%29%28x%5E4%2B4x%5E3y-2x%5E2y%2B4x%5E2y%5E2-4xy%5E2%2By%5E2%29


Now let's finish up...



%28x%2By%29%28x%5E4%2B4x%5E3y-2x%5E2y%2B4x%5E2y%5E2-4xy%5E2%2By%5E2%29 Start with the given expression.


Expand.


Distribute.


Multiply.


x%5E5%2B5x%5E4y-2x%5E3y%2B8x%5E3y%5E2-6x%5E2y%5E2%2Bxy%5E2%2B4y%5E3x%5E2-4y%5E3x%2By%5E3 Now combine like terms.


So %28x%2By%29%28x%5E4%2B4x%5E3y-2x%5E2y%2B4x%5E2y%5E2-4xy%5E2%2By%5E2%29 expands to x%5E5%2B5x%5E4y-2x%5E3y%2B8x%5E3y%5E2-6x%5E2y%5E2%2Bxy%5E2%2B4y%5E3x%5E2-4y%5E3x%2By%5E3.


In other words, .



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Answer:


So %28x%2By%29%28x%5E2%2B2xy-y%29%5E2 fully expands out to x%5E5%2B5x%5E4y-2x%5E3y%2B8x%5E3y%5E2-6x%5E2y%5E2%2Bxy%5E2%2B4y%5E3x%5E2-4y%5E3x%2By%5E3


In other words,



Note: if this problem was more complicated than it should have been, then maybe there's a typo somewhere.

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Believe me this can't be simplified.
It can be expanded and multiplied out.
(x^(2)+2xy-y)^2 becomes x^4+4 x^3 y+4 x^2 y^2-2 x^2 y-4 x y^2+y^2
and then you still have to multiply by (x+y)
and that can become
x^5+(5 x-2) x^3 y+(8 x^2-6 x+1) x y^2+(2 x-1)^2 y^3
or that can become
x^5+5 x^4 y+8 x^3 y^2-2 x^3 y+4 x^2 y^3-6 x^2 y^2-4 x y^3+x y^2+y^3