complicated factoring problem

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• Mar 3rd 2009, 03:08 PM
oblixps
complicated factoring problem
how do you factor this polynomial?

x^(2k+1) - y^(2k+1)

my teacher's answer is:
[x^2 (x^(2k-1) + y^(2k-1))] + [(y^2 - x^2)y^(2k-1)]

what is the reasoning involved to factor something as complicated as this?
• Mar 3rd 2009, 04:20 PM
Abu-Khalil
$x^{2k+1}-y^{2k+1}=x^{2k+1}+x^2y^{2k-1}-x^2y^{2k-1}-y^{2k+1}=x^2\left(x^{2k-1}-y^{2k-1}\right)-\left(x^2-y^2\right)y^{2k-1}$

It's easily to see in this way when you are using induction to prove divisibility.
• Mar 3rd 2009, 07:57 PM
Soroban
Hello, oblixps!

Quote:

How do you factor this polynomial?

. . $x^{2k+1} - y^{2k+1}$

my teacher's answer is: . $\bigg[x^2 \left(x^{2k-1} + y^{2k-1}\right)\bigg] \:{\color{red}+}\: \bigg[\left(y^2 - x^2\right)y^{2k-1}\bigg]$
Wrong! . . . That is not factored!

The difference of two odd powers can always be factored.

$x^{2k+1} - y^{2k+1} \;=\;\bigg(x-y\bigg)\,\bigg(x^{2k} + x^{2k-1}y + x^{2k-2}y^2 + x^{2k-3}y^3 + \hdots + x^2y^{2k-2}$ $+ xy^{2k-1} + y^{2k} \bigg)$