Factor x^8 - y^8 as the product of four polynomials of degree 1, each of which has a positive coefficient of x.

Guest Jun 22, 2021

#1**+1 **

\(\begin{align*} x^8 - y^8 &= (x^4)^2 - (y^4)^2 \\ &= (x^4 - y^4)(x^4 + y^4) \\ &= ((x^2)^2 - (y^2)^2)(x^4 + y^4) \\ &= (x^2 - y^2)(x^2 + y^2)(x^4 + y^4) \\ &= (x-y)(x+y)(x^2 + y^2)(x^4 + y^4) \end{align*}\)

Can you take it from here?

Hint: Manipulate $x^2 + y^2$ into $(x+y)^2 - 2xy$ and do the same thing with $x^4 + y^4.$ You can substitute your equation for $x^2 + y^2$ and do some further manipulation and you are done. If you have degrees $>1,$ you can square root it.

MathProblemSolver101 Jun 22, 2021