Solve 6x^5+12x^3+18x^2

$$6 x ^ { 5 } + 12 x ^ { 3 } + 18 x ^ { 2 }$$

Factor

$$6\left(x+1\right)x^{2}\left(x^{2}-x+3\right)$$

Solution Steps

Factor out 6.

$$6\left(x^{5}+2x^{3}+3x^{2}\right)$$

Consider x^{5}+2x^{3}+3x^{2}. Factor out x^{2}.

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

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

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

Rewrite the complete factored expression. Polynomial x^{2}-x+3 is not factored since it does not have any rational roots.

$$6x^{2}\left(x+1\right)\left(x^{2}-x+3\right)$$

Evaluate

$$6x^{2}\left(x^{3}+2x+3\right)$$