Solve a^6-7a^3b^3-8b^6

$$a ^ { 6 } - 7 a ^ { 3 } b ^ { 3 } - 8 b ^ { 6 }$$

Factor

$$\left(a-2b\right)\left(a+b\right)\left(a^{2}+2ab+4b^{2}\right)\left(a^{2}-ab+b^{2}\right)$$

Solution Steps

Consider a^{6}-7a^{3}b^{3}-8b^{6} as a polynomial over variable a.

$$a^{6}-7b^{3}a^{3}-8b^{6}$$

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{6} and m divides the constant factor -8b^{6}. One such factor is a^{3}-8b^{3}. Factor the polynomial by dividing it by this factor.

$$\left(a^{3}-8b^{3}\right)\left(a^{3}+b^{3}\right)$$

Consider a^{3}-8b^{3}. Rewrite a^{3}-8b^{3} as a^{3}-\left(2b\right)^{3}. The difference of cubes can be factored using the rule: p^{3}-q^{3}=\left(p-q\right)\left(p^{2}+pq+q^{2}\right).

$$\left(a-2b\right)\left(a^{2}+2ab+4b^{2}\right)$$

Consider a^{3}+b^{3}. The sum of cubes can be factored using the rule: p^{3}+q^{3}=\left(p+q\right)\left(p^{2}-pq+q^{2}\right).

$$\left(a+b\right)\left(a^{2}-ab+b^{2}\right)$$

Rewrite the complete factored expression.

$$\left(a-2b\right)\left(a+b\right)\left(a^{2}-ab+b^{2}\right)\left(a^{2}+2ab+4b^{2}\right)$$

Evaluate

$$a^{6}-8b^{6}-7\left(ab\right)^{3}$$