Solve x^4-7x^2-2x+8

$$x ^ { 4 } - 7 x ^ { 2 } - 2 x + 8$$

Factor

$$\left(x-1\right)\left(x+2\right)\left(x^{2}-x-4\right)$$

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 8 and q divides the leading coefficient 1. One such root is -2. Factor the polynomial by dividing it by x+2.

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

Consider x^{3}-2x^{2}-3x+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 1. Factor the polynomial by dividing it by x-1.

$$\left(x-1\right)\left(x^{2}-x-4\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-x-4 is not factored since it does not have any rational roots.

$$\left(x^{2}-x-4\right)\left(x-1\right)\left(x+2\right)$$

Evaluate

$$x^{4}-7x^{2}-2x+8$$