Solve x^4-16x^3+86x^2-176x+120

$$x ^ { 4 } - 16 x ^ { 3 } + 86 x ^ { 2 } - 176 x + 120$$

Factor

$$\left(x-6\right)\left(x-2\right)\left(x^{2}-8x+10\right)$$

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 120 and q divides the leading coefficient 1. One such root is 6. Factor the polynomial by dividing it by x-6.

$$\left(x-6\right)\left(x^{3}-10x^{2}+26x-20\right)$$

Consider x^{3}-10x^{2}+26x-20. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -20 and q divides the leading coefficient 1. One such root is 2. Factor the polynomial by dividing it by x-2.

$$\left(x-2\right)\left(x^{2}-8x+10\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-8x+10 is not factored since it does not have any rational roots.

$$\left(x-6\right)\left(x-2\right)\left(x^{2}-8x+10\right)$$

Evaluate

$$\left(x-6\right)\left(x-2\right)\left(x^{2}-8x+10\right)$$