Solve 6x^7+3x^4-9x^3

$$6 x ^ { 7 } + 3 x ^ { 4 } - 9 x ^ { 3 }$$

Factor

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

Solution Steps

Factor out 3.

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

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

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

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

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

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

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

Evaluate

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