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

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

Factor

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

Solution Steps

Factor out x.

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

Consider 2x^{3}+3x^{2}-1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 2. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2x-1.

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

Consider x^{2}+2x+1. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=x and b=1.

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

Rewrite the complete factored expression.

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

Evaluate

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