Solve x^4-6x^2-7x-6

$$x ^ { 4 } - 6 x ^ { 2 } - 7 x - 6$$

Factor

$$\left(x-3\right)\left(x+2\right)\left(x^{2}+x+1\right)$$

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 -6 and q divides the leading coefficient 1. One such root is 3. Factor the polynomial by dividing it by x-3.

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

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

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

Rewrite the complete factored expression. Polynomial x^{2}+x+1 is not factored since it does not have any rational roots.

$$\left(x-3\right)\left(x^{2}+x+1\right)\left(x+2\right)$$

Evaluate

$$\left(x-3\right)\left(x+2\right)\left(x^{2}+x+1\right)$$