Solve 3a^6-a^5b+3ab^5-b^6

$$3 a ^ { 6 } - a ^ { 5 } b + 3 a b ^ { 5 } - b ^ { 6 }$$

Factor

$$\left(3a-b\right)\left(a+b\right)\left(a^{4}+a^{2}b^{2}-ab^{3}+b^{4}-ba^{3}\right)$$

Solution Steps

Do the grouping 3a^{6}-a^{5}b+3ab^{5}-b^{6}=\left(3a^{6}-a^{5}b\right)+\left(3ab^{5}-b^{6}\right), and factor out a^{5} in the first and b^{5} in the second group.

$$a^{5}\left(3a-b\right)+b^{5}\left(3a-b\right)$$

Factor out common term 3a-b by using distributive property.

$$\left(3a-b\right)\left(a^{5}+b^{5}\right)$$

Consider a^{5}+b^{5}. Consider a^{5}+b^{5} as a polynomial over variable a. Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{5} and m divides the constant factor b^{5}. One such factor is a+b. Factor the polynomial by dividing it by this factor.

$$\left(a+b\right)\left(a^{4}+a^{2}b^{2}-ab^{3}+b^{4}-ba^{3}\right)$$

Rewrite the complete factored expression.

$$\left(a+b\right)\left(3a-b\right)\left(a^{4}+a^{2}b^{2}-ab^{3}+b^{4}-ba^{3}\right)$$

Evaluate

$$3a^{6}+3ab^{5}-b^{6}-ba^{5}$$