Solve 12x^4+4x^3-3x^2-x

$$12 x ^ { 4 } + 4 x ^ { 3 } - 3 x ^ { 2 } - x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(12x^{3}+4x^{2}-3x-1\right)$$

Consider 12x^{3}+4x^{2}-3x-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 12. One such root is -\frac{1}{3}. Factor the polynomial by dividing it by 3x+1.

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

Consider 4x^{2}-1. Rewrite 4x^{2}-1 as \left(2x\right)^{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).

$$\left(2x-1\right)\left(2x+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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