Solve x^5+x^3-30x

$$x ^ { 5 } + x ^ { 3 } - 30 x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(x^{4}+x^{2}-30\right)$$

Consider x^{4}+x^{2}-30. 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 -30. One such factor is x^{2}+6. Factor the polynomial by dividing it by this factor.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}-5,x^{2}+6.

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

Evaluate

$$x\left(x^{4}+x^{2}-30\right)$$