|Apr26-11, 11:59 AM||#1|
Problem 1,v, Ch.1, Spivaks 4th Ed Calculus
Getting back into Physics so doing self study. Having an issue with Spivak's solution to Problem 1, v.
x^n - y^n= (x-y)(x^n-1+x^n-2y+....+xy^n-2+y^n-1
Got all the way to:
How do i get rid of the middle section?
|Apr26-11, 01:44 PM||#2|
Are you trying to show that if you multiply the two factors in your first equation, you get xn - yn?
If so, multiplying the factors on the right gives
xn + xn - 1y - xn - 1y + xn - 2y2 - xn - 2y2 + ... + xyn - 1 - xyn - 1 - yn. Notice that instead of multiplying all of the terms in the larger factor by x, and then repeating the process by multiplying all of the terms by -y, I have instead alternated to get partial products where the exponents add up to the same value.
The first partial product is xn. The next two products come from multiplying x by xn -2y, resulting in xn-1y, and by multiplying -y by xn-1, resulting in -xn - 1y. These two terms add to zero, as do the following pairs of terms.
Doing the multiplication this way makes it clear that all but the first and last terms will drop out.
|algebra, exponent, number properties, proof, spivak|
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