Solve 2x^5-3x^3+x

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

Factor

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

Solution Steps

Factor out x.

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

Consider 2x^{4}-3x^{2}+1. Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 2x^{4} and n divides the constant factor 1. One such factor is 2x^{2}-1. Factor the polynomial by dividing it by this factor.

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

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. Polynomial 2x^{2}-1 is not factored since it does not have any rational roots.

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

Evaluate

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