Solve 6x^5-12x^3+6x

$$6 x ^ { 5 } - 12 x ^ { 3 } + 6 x$$

Factor

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

Solution Steps

Factor out 6.

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

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

$$x\left(x^{4}-2x^{2}+1\right)$$

Consider x^{4}-2x^{2}+1. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor 1. One such factor is x^{2}-1. Factor the polynomial by dividing it by this factor.

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

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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