Solve 343a^3+8b^3+c^3-42abc

$$343 a ^ { 3 } + 8 b ^ { 3 } + c ^ { 3 } - 42 a b c$$

Factor

$$\left(7a+2b+c\right)\left(49a^{2}-14ab-7ac+4b^{2}-2bc+c^{2}\right)$$

Solution Steps

Consider 343a^{3}+8b^{3}+c^{3}-42abc as a polynomial over variable a.

$$343a^{3}-42bca+8b^{3}+c^{3}$$

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power 343a^{3} and n divides the constant factor 8b^{3}+c^{3}. One such factor is 7a+2b+c. Factor the polynomial by dividing it by this factor.

$$\left(7a+2b+c\right)\left(49a^{2}-14ab-7ac+4b^{2}-2bc+c^{2}\right)$$

Evaluate

$$343a^{3}-42abc+8b^{3}+c^{3}$$