Solve f(x)=x^4-18x^3+146x^2-560x+800

$$f ( x ) = x ^ { 4 } - 18 x ^ { 3 } + 146 x ^ { 2 } - 560 x + 800$$

Factor

$$\left(x-4\right)^{2}\left(x^{2}-10x+50\right)$$

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 800 and q divides the leading coefficient 1. One such root is 4. Factor the polynomial by dividing it by x-4.

$$\left(x-4\right)\left(x^{3}-14x^{2}+90x-200\right)$$

Consider x^{3}-14x^{2}+90x-200. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -200 and q divides the leading coefficient 1. One such root is 4. Factor the polynomial by dividing it by x-4.

$$\left(x-4\right)\left(x^{2}-10x+50\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-10x+50 is not factored since it does not have any rational roots.

$$\left(x^{2}-10x+50\right)\left(x-4\right)^{2}$$

Evaluate

$$\left(x-4\right)^{2}\left(x^{2}-10x+50\right)$$