SOLUTION: what is the solution for (x^5-y^5)/(x-y)

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Question 174465: what is the solution for (x^5-y^5)/(x-y)
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
what is the solution for (x^5-y^5)/(x-y)
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Ans: x^4 + x^3y + x^2y^2 + xy^4 + y^4
==========================================
Cheers,
Stan H.

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Edwin's complete solution:
Simplify: %28x%5E5-y%5E5%29%2F%28x-y%29

Most people learn the factorization of x%5EN-y%5EN%29.

But if you forget you can divide that problem out using long division:

              x4  +  x3y  +  x2y2 +  xy3 + y4
      ---------------------------------------
x - y ) x5 + 0x4y + 0x3y2 + 0x2y3 + 0xy4 - y5
        x5 -  x4y
        ---------
              x4y + 0x3y2
              x4y -  x3y2
              -----------
                     x3y2 + 0x2y3
                     x3y2 -  x2y3
                     ------------
                             x2y3 + 0xy4
                             x2y3 -  xy4
                             -----------
                                     xy4 - y5
                                     xy4 - y5
                                     --------
                                            0

So the answer is 

x4  +  x3y  +  x2y2 +  xy3 + y4


However if you memorize how to factor x%5EN+-+y%5EN, i.e.,



then you will know how to factor x%5E5-y%5E5. Then to simplify
the expression

%28x%5E5-y%5E5%29%2F%28x-y%29

you will just factor the numerator according to the rule above:

%28%28x-y%29%28x%5E4%2Bx%5E3y%2Bx%5E2y%5E2%2Bxy%5E3%2By%5E4%29%29%2F%28x-y%29



x%5E4%2Bx%5E3y%2Bx%5E2y%5E2%2Bxy%5E3%2By%5E4%29.

One thing to remember though, is that although that factorization
method holds for all values of N, it only factors COMPLETELY when
N is an ODD positive integer.  When N is EVEN, you should always 
begin by factoring it as the difference of two squares.  

Edwin