Solve x^3-27x-54

$$x ^ { 3 } - 27 x - 54$$

Factor

$$\left(x-6\right)\left(x+3\right)^{2}$$

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 -54 and q divides the leading coefficient 1. One such root is 6. Factor the polynomial by dividing it by x-6.

$$\left(x-6\right)\left(x^{2}+6x+9\right)$$

Consider x^{2}+6x+9. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=x and b=3.

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

Rewrite the complete factored expression.

$$\left(x-6\right)\left(x+3\right)^{2}$$

Evaluate

$$\left(x-6\right)\left(x+3\right)^{2}$$