Solve 3x^5+3x^4-21x^3+6x^2

$$3 x ^ { 5 } + 3 x ^ { 4 } - 21 x ^ { 3 } + 6 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(x^{5}+x^{4}-7x^{3}+2x^{2}\right)$$

Consider x^{5}+x^{4}-7x^{3}+2x^{2}. Factor out x^{2}.

$$x^{2}\left(x^{3}+x^{2}-7x+2\right)$$

Consider x^{3}+x^{2}-7x+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 1. One such root is 2. Factor the polynomial by dividing it by x-2.

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

Rewrite the complete factored expression. Polynomial x^{2}+3x-1 is not factored since it does not have any rational roots.

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

Evaluate

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