Solve a^5+a^4b-a^3-a^2b

$$a ^ { 5 } + a ^ { 4 } b - a ^ { 3 } - a ^ { 2 } b$$

Factor

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

Solution Steps

Factor out a^{2}.

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

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

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

Factor out common term a+b by using distributive property.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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