Solve 2x^4-x^3+2x-1

$$2 x ^ { 4 } - x ^ { 3 } + 2 x - 1$$

Factor

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

Solution Steps

Do the grouping 2x^{4}-x^{3}+2x-1=\left(2x^{4}-x^{3}\right)+\left(2x-1\right), and factor out x^{3} in 2x^{4}-x^{3}.

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

Factor out common term 2x-1 by using distributive property.

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

Consider x^{3}+1. Rewrite x^{3}+1 as x^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

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

Rewrite the complete factored expression. Polynomial x^{2}-x+1 is not factored since it does not have any rational roots.

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

Evaluate

$$2x^{4}-x^{3}+2x-1$$