- #1
Magicalz
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Homework Statement
Prove the following:
[tex] x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1}) [/tex]
The Attempt at a Solution
Ugh I just tried to distribute the right part:
[tex]
\begin{equation*}
\begin{split}
\x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1}) \\
&= \(x - y)x^{n-1} + (x - y)x^{n-2}y + ... + (x - y)xy^{n-2} + (x - y)y^{n-1} \\
&= x^n - x^{n-1}y + x^{n-1}y - x^{n-2}y^2 + ... + x^2y^{n-2} - xy^{n-1} + xy^n-1 - y^n
\end{split}
\end{equation*}
[/tex]
well the terms that are visible (that are written down) really do cancel out (except of course x^n and y^n) however I guess what I'm having trouble grasping with is what constitutes the "proof" part of this...I mean yes it seems likely that the terms inbetween x^n and y^n are cancelling each other out, but can we be absolutely sure? the dots ... mean continue the pattern,but I guess what I'm wondering is,is what I'm doing by distributing the right side constituting the "proof"? Any help is appreciated thankyou.
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