Solve a^5-a^3-12a=

$$a ^ { 5 } - a ^ { 3 } - 12 a =$$

Factor

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

Solution Steps

Factor out a.

$$a\left(a^{4}-a^{2}-12\right)$$

Consider a^{4}-a^{2}-12. Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{4} and m divides the constant factor -12. One such factor is a^{2}-4. Factor the polynomial by dividing it by this factor.

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

Consider a^{2}-4. Rewrite a^{2}-4 as a^{2}-2^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a-2\right)\left(a+2\right)$$

Rewrite the complete factored expression. Polynomial a^{2}+3 is not factored since it does not have any rational roots.

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

Evaluate

$$a\left(a^{4}-a^{2}-12\right)$$