Solve a^3-12a-16

$${ a }^{ 3 } -12a-16$$

Factor

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

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -16 and q divides the leading coefficient 1. One such root is 4. Factor the polynomial by dividing it by a-4.

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

Consider a^{2}+4a+4. Use the perfect square formula, p^{2}+2pq+q^{2}=\left(p+q\right)^{2}, where p=a and q=2.

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

Rewrite the complete factored expression.

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

Evaluate

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