Solve a^{4}+ab^{3}+3a^{3}b+3a^{2}b^{2}+a^{3}b+b^4+3a^2b^2+3ab^3

$$a ^ { 4 } + a b ^ { 3 } + 3 a ^ { 3 } b + 3 a ^ { 2 } b ^ { 2 } + a ^ { 3 } b + b ^ { 4 } + 3 a ^ { 2 } b ^ { 2 } + 3 a b ^ { 3 }$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$a^{4}+6a^{2}b^{2}+4ab^{3}+b^{4}+4ba^{3}$$

Consider a^{4}+6a^{2}b^{2}+4ab^{3}+b^{4}+4ba^{3} as a polynomial over variable a.

$$a^{4}+4ba^{3}+6b^{2}a^{2}+4b^{3}a+b^{4}$$

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

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

Consider a^{3}+3ab^{2}+b^{3}+3ba^{2}. Use the binomial cube formula, p^{3}+3p^{2}q+3pq^{2}+q^{3}=\left(p+q\right)^{3}, where p=a and q=b.

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

Rewrite the complete factored expression.

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

Evaluate

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