Solve q^6-6q^3-7

$$q ^ { 6 } - 6 q ^ { 3 } - 7$$

Factor

$$\left(q+1\right)\left(q^{2}-q+1\right)\left(q^{3}-7\right)$$

Solution Steps

Find one factor of the form q^{k}+m, where q^{k} divides the monomial with the highest power q^{6} and m divides the constant factor -7. One such factor is q^{3}-7. Factor the polynomial by dividing it by this factor.

$$\left(q^{3}-7\right)\left(q^{3}+1\right)$$

Consider q^{3}+1. Rewrite q^{3}+1 as q^{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).

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: q^{3}-7,q^{2}-q+1.

$$\left(q^{3}-7\right)\left(q^{2}-q+1\right)\left(q+1\right)$$

Evaluate

$$q^{6}-6q^{3}-7$$