Solve 3x^4-14x^3-6x^2+6x-5

$$3 x ^ { 4 } - 14 x ^ { 3 } - 6 x ^ { 2 } + 6 x - 5$$

Factor

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

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 -5 and q divides the leading coefficient 3. One such root is 5. Factor the polynomial by dividing it by x-5.

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

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

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

Rewrite the complete factored expression. Polynomial 3x^{2}-2x+1 is not factored since it does not have any rational roots.

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

Evaluate

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