Solve 6x^5-12x^3-18x

$$6 x ^ { 5 } - 12 x ^ { 3 } - 18 x$$

Factor

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

Solution Steps

Factor out 6.

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

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

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

Consider x^{4}-2x^{2}-3. 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 -3. One such factor is x^{2}-3. Factor the polynomial by dividing it by this factor.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}-3,x^{2}+1.

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

Evaluate

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