How do you factor x^5-y^5?

(using only real coefficients)

Jul 21, 2017

${x}^{5} - {y}^{5} = \left(x - y\right) \left({x}^{4} + {x}^{3} y + {x}^{2} {y}^{2} + x {y}^{3} + {y}^{4}\right)$

Explanation:

We know that :

${x}^{n} - {y}^{n} = \left(x - y\right) \left({x}^{n - 1} + {x}^{n - 2} y + \ldots + x {y}^{n - 2} + {y}^{n - 1}\right)$

So let's use this :

${x}^{5} - {y}^{5} = \left(x - y\right) \left({x}^{4} + {x}^{3} y + {x}^{2} {y}^{2} + x {y}^{3} + {y}^{4}\right)$

Jul 21, 2017

${x}^{5} - {y}^{5} = \left(x - y\right) \left({x}^{2} + \left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) x y + {y}^{2}\right) \left({x}^{2} + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right) x y + {y}^{2}\right)$

Explanation:

Given:

${x}^{5} - {y}^{5}$

First note that if $x = y$ then ${x}^{5} - {y}^{5} = 0$. Hence we can deduce that $\left(x - y\right)$ is a factor:

${x}^{5} - {y}^{5} = \left(x - y\right) \left({x}^{4} + {x}^{3} y + {x}^{2} {y}^{2} + x {y}^{3} + {y}^{4}\right)$

We can factor the remaining quartic by making use of its symmetry, expressing it in terms of a quadratic in $\left(\frac{x}{y} + \frac{y}{x}\right)$ as follows:

Note that:

${\left(\frac{x}{y} + \frac{y}{x}\right)}^{2} = {x}^{2} / {y}^{2} + 2 + {y}^{2} / {x}^{2}$

So we find:

${x}^{4} + {x}^{3} y + {x}^{2} {y}^{2} + x {y}^{3} + {y}^{4}$

$= {x}^{2} {y}^{2} \left({x}^{2} / {y}^{2} + \frac{x}{y} + 1 + \frac{y}{x} + {y}^{2} / {x}^{2}\right)$

$= {x}^{2} {y}^{2} \left({\left(\frac{x}{y} + \frac{y}{x}\right)}^{2} + \left(\frac{x}{y} + \frac{y}{x}\right) - 1\right)$

$= {x}^{2} {y}^{2} \left({\left(\frac{x}{y} + \frac{y}{x}\right)}^{2} + \left(\frac{x}{y} + \frac{y}{x}\right) + \frac{1}{4} - \frac{5}{4}\right)$

$= {x}^{2} {y}^{2} \left({\left(\left(\frac{x}{y} + \frac{y}{x}\right) + \frac{1}{2}\right)}^{2} - {\left(\frac{\sqrt{5}}{2}\right)}^{2}\right)$

= x^2y^2((x/y+y/x)+1/2)-sqrt(5)/2)((x/y+y/x)+1/2)+sqrt(5)/2)

$= {x}^{2} {y}^{2} \left(\frac{x}{y} + \left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) + \frac{y}{x}\right) \left(\frac{x}{y} + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right) + \frac{y}{x}\right)$

$= \left({x}^{2} + \left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) x y + {y}^{2}\right) \left({x}^{2} + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right) x y + {y}^{2}\right)$

So putting it all together, we have:

${x}^{5} - {y}^{5} = \left(x - y\right) \left({x}^{2} + \left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) x y + {y}^{2}\right) \left({x}^{2} + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right) x y + {y}^{2}\right)$