Solve left(a-bright)^3+left(a+bright)^3+3left(a+bright)left(a-bright)^2+3left(a-bright)left(a+bright)^2

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

Evaluate

$$8a^{3}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(a-b\right)^{3}.

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

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(a+b\right)^{3}.

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

Combine a^{3} and a^{3} to get 2a^{3}.

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

Combine -3a^{2}b and 3a^{2}b to get 0.

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

Combine 3ab^{2} and 3ab^{2} to get 6ab^{2}.

$$2a^{3}+6ab^{2}-b^{3}+b^{3}+3\left(a+b\right)\left(a-b\right)^{2}+3\left(a-b\right)\left(a+b\right)^{2}$$

Combine -b^{3} and b^{3} to get 0.

$$2a^{3}+6ab^{2}+3\left(a+b\right)\left(a-b\right)^{2}+3\left(a-b\right)\left(a+b\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$2a^{3}+6ab^{2}+3\left(a+b\right)\left(a^{2}-2ab+b^{2}\right)+3\left(a-b\right)\left(a+b\right)^{2}$$

Use the distributive property to multiply 3 by a+b.

$$2a^{3}+6ab^{2}+\left(3a+3b\right)\left(a^{2}-2ab+b^{2}\right)+3\left(a-b\right)\left(a+b\right)^{2}$$

Use the distributive property to multiply 3a+3b by a^{2}-2ab+b^{2} and combine like terms.

$$2a^{3}+6ab^{2}+3a^{3}-3a^{2}b-3ab^{2}+3b^{3}+3\left(a-b\right)\left(a+b\right)^{2}$$

Combine 2a^{3} and 3a^{3} to get 5a^{3}.

$$5a^{3}+6ab^{2}-3a^{2}b-3ab^{2}+3b^{3}+3\left(a-b\right)\left(a+b\right)^{2}$$

Combine 6ab^{2} and -3ab^{2} to get 3ab^{2}.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3\left(a-b\right)\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+b\right)^{2}.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3\left(a-b\right)\left(a^{2}+2ab+b^{2}\right)$$

Use the distributive property to multiply 3 by a-b.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+\left(3a-3b\right)\left(a^{2}+2ab+b^{2}\right)$$

Use the distributive property to multiply 3a-3b by a^{2}+2ab+b^{2} and combine like terms.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3a^{3}+3a^{2}b-3ab^{2}-3b^{3}$$

Combine 5a^{3} and 3a^{3} to get 8a^{3}.

$$8a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3a^{2}b-3ab^{2}-3b^{3}$$

Combine -3a^{2}b and 3a^{2}b to get 0.

$$8a^{3}+3ab^{2}+3b^{3}-3ab^{2}-3b^{3}$$

Combine 3ab^{2} and -3ab^{2} to get 0.

$$8a^{3}+3b^{3}-3b^{3}$$

Combine 3b^{3} and -3b^{3} to get 0.

$$8a^{3}$$

Expand

$$8a^{3}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(a-b\right)^{3}.

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

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(a+b\right)^{3}.

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

Combine a^{3} and a^{3} to get 2a^{3}.

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

Combine -3a^{2}b and 3a^{2}b to get 0.

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

Combine 3ab^{2} and 3ab^{2} to get 6ab^{2}.

$$2a^{3}+6ab^{2}-b^{3}+b^{3}+3\left(a+b\right)\left(a-b\right)^{2}+3\left(a-b\right)\left(a+b\right)^{2}$$

Combine -b^{3} and b^{3} to get 0.

$$2a^{3}+6ab^{2}+3\left(a+b\right)\left(a-b\right)^{2}+3\left(a-b\right)\left(a+b\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$2a^{3}+6ab^{2}+3\left(a+b\right)\left(a^{2}-2ab+b^{2}\right)+3\left(a-b\right)\left(a+b\right)^{2}$$

Use the distributive property to multiply 3 by a+b.

$$2a^{3}+6ab^{2}+\left(3a+3b\right)\left(a^{2}-2ab+b^{2}\right)+3\left(a-b\right)\left(a+b\right)^{2}$$

Use the distributive property to multiply 3a+3b by a^{2}-2ab+b^{2} and combine like terms.

$$2a^{3}+6ab^{2}+3a^{3}-3a^{2}b-3ab^{2}+3b^{3}+3\left(a-b\right)\left(a+b\right)^{2}$$

Combine 2a^{3} and 3a^{3} to get 5a^{3}.

$$5a^{3}+6ab^{2}-3a^{2}b-3ab^{2}+3b^{3}+3\left(a-b\right)\left(a+b\right)^{2}$$

Combine 6ab^{2} and -3ab^{2} to get 3ab^{2}.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3\left(a-b\right)\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+b\right)^{2}.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3\left(a-b\right)\left(a^{2}+2ab+b^{2}\right)$$

Use the distributive property to multiply 3 by a-b.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+\left(3a-3b\right)\left(a^{2}+2ab+b^{2}\right)$$

Use the distributive property to multiply 3a-3b by a^{2}+2ab+b^{2} and combine like terms.

$$5a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3a^{3}+3a^{2}b-3ab^{2}-3b^{3}$$

Combine 5a^{3} and 3a^{3} to get 8a^{3}.

$$8a^{3}+3ab^{2}-3a^{2}b+3b^{3}+3a^{2}b-3ab^{2}-3b^{3}$$

Combine -3a^{2}b and 3a^{2}b to get 0.

$$8a^{3}+3ab^{2}+3b^{3}-3ab^{2}-3b^{3}$$

Combine 3ab^{2} and -3ab^{2} to get 0.

$$8a^{3}+3b^{3}-3b^{3}$$

Combine 3b^{3} and -3b^{3} to get 0.

$$8a^{3}$$