Solve x^5-x^3-2x

$${ x }^{ 5 } - { x }^{ 3 } -2x$$

Factor

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

Solution Steps

Factor out x.

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

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

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

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

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

Evaluate

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