Solve 3x^6-3x^3-6

$$3 x ^ { 6 } - 3 x ^ { 3 } - 6$$

Factor

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

Solution Steps

Factor out 3.

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

Consider x^{6}-x^{3}-2. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{6} and m divides the constant factor -2. One such factor is x^{3}-2. Factor the polynomial by dividing it by this factor.

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

Consider x^{3}+1. Rewrite x^{3}+1 as x^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{3}-2,x^{2}-x+1.

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

Evaluate

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