Solve 3x^4-11x^3+59x^2-81x-34

$$3 x ^ { 4 } - 11 x ^ { 3 } + 59 x ^ { 2 } - 81 x - 34$$

Factor

$$\left(x-2\right)\left(3x+1\right)\left(x^{2}-2x+17\right)$$

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 -34 and q divides the leading coefficient 3. One such root is -\frac{1}{3}. Factor the polynomial by dividing it by 3x+1.

$$\left(3x+1\right)\left(x^{3}-4x^{2}+21x-34\right)$$

Consider x^{3}-4x^{2}+21x-34. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -34 and q divides the leading coefficient 1. One such root is 2. Factor the polynomial by dividing it by x-2.

$$\left(x-2\right)\left(x^{2}-2x+17\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-2x+17 is not factored since it does not have any rational roots.

$$\left(x-2\right)\left(3x+1\right)\left(x^{2}-2x+17\right)$$

Evaluate

$$\left(x-2\right)\left(3x+1\right)\left(x^{2}-2x+17\right)$$