Solve a^{2}x^{3}-x^{3}b^{2}-a^2y^3+y^3b^2=

$$a ^ { 2 } x ^ { 3 } - x ^ { 3 } b ^ { 2 } - a ^ { 2 } y ^ { 3 } + y ^ { 3 } b ^ { 2 } =$$

Factor

$$\left(a-b\right)\left(x-y\right)\left(a+b\right)\left(x^{2}+xy+y^{2}\right)$$

Solution Steps

Do the grouping a^{2}x^{3}-x^{3}b^{2}-a^{2}y^{3}+y^{3}b^{2}=\left(a^{2}x^{3}-x^{3}b^{2}\right)+\left(-a^{2}y^{3}+y^{3}b^{2}\right), and factor out x^{3} in the first and -y^{3} in the second group.

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

Factor out common term a^{2}-b^{2} by using distributive property.

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

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

$$\left(a-b\right)\left(a+b\right)$$

Consider x^{3}-y^{3}. The difference of cubes can be factored using the rule: p^{3}-q^{3}=\left(p-q\right)\left(p^{2}+pq+q^{2}\right).

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

Rewrite the complete factored expression.

$$\left(a-b\right)\left(a+b\right)\left(x-y\right)\left(x^{2}+xy+y^{2}\right)$$

Evaluate

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