Solve 6x^4-11x^3-x^2-4

$$6 x ^ { 4 } - 11 x ^ { 3 } - x ^ { 2 } - 4$$

Factor

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

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 6. One such root is -\frac{2}{3}. Factor the polynomial by dividing it by 3x+2.

$$\left(3x+2\right)\left(2x^{3}-5x^{2}+3x-2\right)$$

Consider 2x^{3}-5x^{2}+3x-2. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 2. One such root is 2. Factor the polynomial by dividing it by x-2.

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

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

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

Evaluate

$$6x^{4}-11x^{3}-x^{2}-4$$