Solve x^5-9x^4+31x^3-49x^2+36x-10

$${ x }^{ 5 } -9 { x }^{ 4 } +31 { x }^{ 3 } -49 { x }^{ 2 } +36x-10$$

Factor

$$\left(x^{2}-6x+10\right)\left(x-1\right)^{3}$$

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 -10 and q divides the leading coefficient 1. One such root is 1. Factor the polynomial by dividing it by x-1.

$$\left(x-1\right)\left(x^{4}-8x^{3}+23x^{2}-26x+10\right)$$

Consider x^{4}-8x^{3}+23x^{2}-26x+10. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 10 and q divides the leading coefficient 1. One such root is 1. Factor the polynomial by dividing it by x-1.

$$\left(x-1\right)\left(x^{3}-7x^{2}+16x-10\right)$$

Consider x^{3}-7x^{2}+16x-10. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -10 and q divides the leading coefficient 1. One such root is 1. Factor the polynomial by dividing it by x-1.

$$\left(x-1\right)\left(x^{2}-6x+10\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-6x+10 is not factored since it does not have any rational roots.

$$\left(x^{2}-6x+10\right)\left(x-1\right)^{3}$$

Evaluate

$$\left(x^{2}-6x+10\right)\left(x-1\right)^{3}$$