Solve =x^5-6x^4+13x^3-12x^2+4x

$$= x ^ { 5 } - 6 x ^ { 4 } + 13 x ^ { 3 } - 12 x ^ { 2 } + 4 x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(x^{4}-6x^{3}+13x^{2}-12x+4\right)$$

Consider x^{4}-6x^{3}+13x^{2}-12x+4. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. One such root is 2. Factor the polynomial by dividing it by x-2.

$$\left(x-2\right)\left(x^{3}-4x^{2}+5x-2\right)$$

Consider x^{3}-4x^{2}+5x-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 2. Factor the polynomial by dividing it by x-2.

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

Consider x^{2}-2x+1. Use the perfect square formula, a^{2}-2ab+b^{2}=\left(a-b\right)^{2}, where a=x and b=1.

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

Rewrite the complete factored expression.

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

Evaluate

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