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

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

Factor

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

Solution Steps

Factor out a^{2}.

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

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

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

Factor out common term -a+2 by using distributive property.

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

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.

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

Evaluate

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