Solve 20x^5+30x^4-4x^3-6x^2

$$20 x ^ { 5 } + 30 x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 }$$

Factor

$$2\left(2x+3\right)x^{2}\left(5x^{2}-1\right)$$

Solution Steps

Factor out 2.

$$2\left(10x^{5}+15x^{4}-2x^{3}-3x^{2}\right)$$

Consider 10x^{5}+15x^{4}-2x^{3}-3x^{2}. Factor out x^{2}.

$$x^{2}\left(10x^{3}+15x^{2}-2x-3\right)$$

Consider 10x^{3}+15x^{2}-2x-3. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -3 and q divides the leading coefficient 10. One such root is -\frac{3}{2}. Factor the polynomial by dividing it by 2x+3.

$$\left(2x+3\right)\left(5x^{2}-1\right)$$

Rewrite the complete factored expression. Polynomial 5x^{2}-1 is not factored since it does not have any rational roots.

$$2x^{2}\left(2x+3\right)\left(5x^{2}-1\right)$$

Evaluate

$$2\left(2x+3\right)x^{2}\left(5x^{2}-1\right)$$