Solve p^3q^3-p^2q^2-pq+1

$$p ^ { 3 } q ^ { 3 } - p ^ { 2 } q ^ { 2 } - p q + 1$$

Factor

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

Solution Steps

Do the grouping p^{3}q^{3}-p^{2}q^{2}-pq+1=\left(p^{3}q^{3}-p^{2}q^{2}\right)+\left(-pq+1\right), and factor out p^{2}q^{2} in the first and -1 in the second group.

$$p^{2}q^{2}\left(pq-1\right)-\left(pq-1\right)$$

Factor out common term pq-1 by using distributive property.

$$\left(pq-1\right)\left(p^{2}q^{2}-1\right)$$

Consider p^{2}q^{2}-1. Rewrite p^{2}q^{2}-1 as \left(pq\right)^{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).

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

Rewrite the complete factored expression.

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

Evaluate

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