Solve 5x^4-31x^3-49x^2+591x-116

$$5 { x }^{ 4 } -31 { x }^{ 3 } -49 { x }^{ 2 } +591x-116$$

Factor

$$\left(5x-1\right)\left(x+4\right)\left(x^{2}-10x+29\right)$$

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 -116 and q divides the leading coefficient 5. One such root is \frac{1}{5}. Factor the polynomial by dividing it by 5x-1.

$$\left(5x-1\right)\left(x^{3}-6x^{2}-11x+116\right)$$

Consider x^{3}-6x^{2}-11x+116. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 116 and q divides the leading coefficient 1. One such root is -4. Factor the polynomial by dividing it by x+4.

$$\left(x+4\right)\left(x^{2}-10x+29\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-10x+29 is not factored since it does not have any rational roots.

$$\left(5x-1\right)\left(x+4\right)\left(x^{2}-10x+29\right)$$

Evaluate

$$\left(5x-1\right)\left(x+4\right)\left(x^{2}-10x+29\right)$$