Solve y^4-y^3+y^2-3y+2

$$y ^ { 4 } - y ^ { 3 } + y ^ { 2 } - 3 y + 2$$

Factor

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

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

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

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

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

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

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

Evaluate

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