Solve x^3y^2-x^3-xy^2+x

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

Factor

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

Solution Steps

Factor out x.

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

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

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

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

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

Consider y^{2}-1. Rewrite y^{2}-1 as y^{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).

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

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.

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

Evaluate

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