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?
Hello, oblixps!
How do you factor this polynomial?
. . $\displaystyle x^{2k+1} - y^{2k+1}$
my teacher's answer is: .$\displaystyle \bigg[x^2 \left(x^{2k-1} + y^{2k-1}\right)\bigg] \:{\color{red}+}\: \bigg[\left(y^2 - x^2\right)y^{2k-1}\bigg]$
Wrong! . . . That is not factored!
The difference of two odd powers can always be factored.
$\displaystyle x^{2k+1} - y^{2k+1} \;=\;\bigg(x-y\bigg)\,\bigg(x^{2k} + x^{2k-1}y + x^{2k-2}y^2 + x^{2k-3}y^3 + \hdots + x^2y^{2k-2}$ $\displaystyle + xy^{2k-1} + y^{2k} \bigg)$