Solve x^5y-x^3y-x^5+x^3

$${ x }^{ 5 } y- { x }^{ 3 } y- { x }^{ 5 } + { x }^{ 3 }$$

Factor

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

Solution Steps

Factor out x^{3}.

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

Consider x^{2}y-y-x^{2}+1. Do the grouping x^{2}y-y-x^{2}+1=\left(x^{2}y-y\right)+\left(-x^{2}+1\right), and factor out y in the first and -1 in the second group.

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

Factor out common term x^{2}-1 by using distributive property.

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

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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