# How do you factor the expression x^3 - x^2y - y^3 + xy^2?

Jul 1, 2016

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

#### Explanation:

${x}^{3} - {x}^{2} y - {y}^{3} + x {y}^{2} = {y}^{3} \left({\left(\frac{x}{y}\right)}^{3} - {\left(\frac{x}{y}\right)}^{2} + \left(\frac{x}{y}\right) - 1\right)$

but

${z}^{3} - {z}^{2} + z - 1 = 0$ has a root $z = 1$

making

${z}^{3} - {z}^{2} + z - 1 = \left(z - 1\right) \left(b {z}^{2} + c z + d\right)$

equating the coefficients we find

{ (d-1 = 0), (c - d + 1= 0),( b - c -1= 0), (1 - b = 0) :}

solving for $b , c , d$

$\left(b = 1 , c = 0 , d = 1\right)$

so

${z}^{3} - {z}^{2} + z - 1 = \left(z - 1\right) \left({z}^{2} + 1\right)$

and finally

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