Solve 1/4a^5-1/2a^3-a^2

$$\frac { 1 } { 4 } a ^ { 5 } - \frac { 1 } { 2 } a ^ { 3 } - a ^ { 2 }$$

Factor

$$\frac{\left(a-2\right)a^{2}\left(a^{2}+2a+2\right)}{4}$$

Solution Steps

Factor out \frac{1}{4}.

$$\frac{a^{5}-2a^{3}-4a^{2}}{4}$$

Consider a^{5}-2a^{3}-4a^{2}. Factor out a^{2}.

$$a^{2}\left(a^{3}-2a-4\right)$$

Consider a^{3}-2a-4. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 1. One such root is 2. Factor the polynomial by dividing it by a-2.

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

Rewrite the complete factored expression. Polynomial a^{2}+2a+2 is not factored since it does not have any rational roots.

$$\frac{a^{2}\left(a-2\right)\left(a^{2}+2a+2\right)}{4}$$

Evaluate

$$\frac{a^{2}\left(a^{3}-2a-4\right)}{4}$$