Copyright 2010, Randy Strauss, All Rights Reserved
Answer: just do the long division and see how it comes out
Just look at the first number, x5
1. x * x4 = x5, so (x-y) goes into x5 how many times? x4:
2. So multiply (x-y) * x4 and subtract this product from x5-y5.
3. You're left with the remainder (after subtracting).
4. Then ask, how many times does x-y go into what was left?
This is the same kind of question as #1, so repeat steps 1, 2, and 3:
Arrange it just like long division, but you're putting x into a sum of powers of x & y instead of a number into powers of 10.
Here's step 1- put the x4 into the answer on top, multiply x-y * x4 and get x5 - yx4, and subract it from x5-y5:
x4
---------------------------------------------------------------
x-y | x5 + - y5
__x5_-__x4y__ -- subtract this from above
x4y + - y5 -- and this is what's left
now we try to put x-y into the term on the left: x4y:
x4 + x3y + x2y2
---------------------------------------------------------------
x-y | x4y + - y5
_x4y_-_x3y2__ -- multiply x3y*(x-y) and subtract it from above
x3y2 - y5 -- this is the new remainder
_x3y2_-_x2y3_ -- mult x2y2 * (x-y) and subtract it from above
x2y3 - y5 -- the new remainder
And we keep going like this. We divide x-y into the new remainder:
x4 + x3y + x2y2 + x y3 + y4
---------------------------------------------------------------
x-y | x2y3 - y5 -- the new remainder
_x2y3_-_xy4_ -- subtract (x-y)(x y3) from last remainder
xy4 - y5 -- new remainder, x-y goes y4 into this
_xy4 - y5 -- wow! it's the same- it went in evenly!
x-y always goes into xn - yn evenly...
You can see from the pattern.