Solve 12u^5-30u^4+12u^3+12u^2-6u

$$12 u ^ { 5 } - 30 u ^ { 4 } + 12 u ^ { 3 } + 12 u ^ { 2 } - 6 u$$

Factor

$$6u\left(u-1\right)\left(2u-1\right)\left(u^{2}-u-1\right)$$

Solution Steps

Factor out 6.

$$6\left(2u^{5}-5u^{4}+2u^{3}+2u^{2}-u\right)$$

Consider 2u^{5}-5u^{4}+2u^{3}+2u^{2}-u. Factor out u.

$$u\left(2u^{4}-5u^{3}+2u^{2}+2u-1\right)$$

Consider 2u^{4}-5u^{3}+2u^{2}+2u-1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 2. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2u-1.

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

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

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

Rewrite the complete factored expression. Polynomial u^{2}-u-1 is not factored since it does not have any rational roots.

$$6u\left(2u-1\right)\left(u-1\right)\left(u^{2}-u-1\right)$$

Evaluate

$$6u\left(u-1\right)\left(2u-1\right)\left(u^{2}-u-1\right)$$