Solve a^3+8b^3+64c^3-24abc

$$a ^ { 3 } + 8 b ^ { 3 } + 64 c ^ { 3 } - 24 a b c$$

Factor

$$\left(a+2b+4c\right)\left(a^{2}-2ab-4ac+4b^{2}-8bc+16c^{2}\right)$$

Solution Steps

Consider a^{3}+8b^{3}+64c^{3}-24abc as a polynomial over variable a.

$$a^{3}-24bca+8b^{3}+64c^{3}$$

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{3} and m divides the constant factor 8b^{3}+64c^{3}. One such factor is a+2b+4c. Factor the polynomial by dividing it by this factor.

$$\left(a+2b+4c\right)\left(a^{2}-2ab-4ac+4b^{2}-8bc+16c^{2}\right)$$

Evaluate

$$a^{3}-24abc+8b^{3}+64c^{3}$$