Solve 2x^4-9x^3+11x^2-7x+3

$$2 x ^ { 4 } - 9 x ^ { 3 } + 11 x ^ { 2 } - 7 x + 3$$

Factor

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

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 3 and q divides the leading coefficient 2. One such root is 3. Factor the polynomial by dividing it by x-3.

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

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

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

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

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

Evaluate

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