Solve 3a^2b^2left(a^2-b^2right)-{left({a}^{2}-{b}^{2}right)}^{3}+left({a}^{3}-{b}^{3}right)left({a}^{3}+{b}^{3}right)-6left({a}^2b+a{b}^2right)left({a}^2b-a{b}^2right)

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

Evaluate

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Solution Steps

Use the distributive property to multiply 3a^{2}b^{2} by a^{2}-b^{2}.

$$3b^{2}a^{4}-3a^{2}b^{4}-\left(a^{2}-b^{2}\right)^{3}+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

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

$$3b^{2}a^{4}-3a^{2}b^{4}-\left(\left(a^{2}\right)^{3}-3\left(a^{2}\right)^{2}b^{2}+3a^{2}\left(b^{2}\right)^{2}-\left(b^{2}\right)^{3}\right)+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$3b^{2}a^{4}-3a^{2}b^{4}-\left(a^{6}-3\left(a^{2}\right)^{2}b^{2}+3a^{2}\left(b^{2}\right)^{2}-\left(b^{2}\right)^{3}\right)+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$3b^{2}a^{4}-3a^{2}b^{4}-\left(a^{6}-3a^{4}b^{2}+3a^{2}\left(b^{2}\right)^{2}-\left(b^{2}\right)^{3}\right)+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$3b^{2}a^{4}-3a^{2}b^{4}-\left(a^{6}-3a^{4}b^{2}+3a^{2}b^{4}-\left(b^{2}\right)^{3}\right)+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$3b^{2}a^{4}-3a^{2}b^{4}-\left(a^{6}-3a^{4}b^{2}+3a^{2}b^{4}-b^{6}\right)+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

To find the opposite of a^{6}-3a^{4}b^{2}+3a^{2}b^{4}-b^{6}, find the opposite of each term.

$$3b^{2}a^{4}-3a^{2}b^{4}-a^{6}+3a^{4}b^{2}-3a^{2}b^{4}+b^{6}+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

Combine 3b^{2}a^{4} and 3a^{4}b^{2} to get 6b^{2}a^{4}.

$$6b^{2}a^{4}-3a^{2}b^{4}-a^{6}-3a^{2}b^{4}+b^{6}+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

Combine -3a^{2}b^{4} and -3a^{2}b^{4} to get -6a^{2}b^{4}.

$$6b^{2}a^{4}-6a^{2}b^{4}-a^{6}+b^{6}+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

Consider \left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$6b^{2}a^{4}-6a^{2}b^{4}-a^{6}+b^{6}+\left(a^{3}\right)^{2}-\left(b^{3}\right)^{2}-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$6b^{2}a^{4}-6a^{2}b^{4}-a^{6}+b^{6}+a^{6}-\left(b^{3}\right)^{2}-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$6b^{2}a^{4}-6a^{2}b^{4}-a^{6}+b^{6}+a^{6}-b^{6}-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

Combine -a^{6} and a^{6} to get 0.

$$6b^{2}a^{4}-6a^{2}b^{4}+b^{6}-b^{6}-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

Combine b^{6} and -b^{6} to get 0.

$$6b^{2}a^{4}-6a^{2}b^{4}-6\left(a^{2}b+ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

Use the distributive property to multiply -6 by a^{2}b+ab^{2}.

$$6b^{2}a^{4}-6a^{2}b^{4}+\left(-6a^{2}b-6ab^{2}\right)\left(a^{2}b-ab^{2}\right)$$

Use the distributive property to multiply -6a^{2}b-6ab^{2} by a^{2}b-ab^{2} and combine like terms.

$$6b^{2}a^{4}-6a^{2}b^{4}-6a^{4}b^{2}+6a^{2}b^{4}$$

Combine 6b^{2}a^{4} and -6a^{4}b^{2} to get 0.

$$-6a^{2}b^{4}+6a^{2}b^{4}$$

Combine -6a^{2}b^{4} and 6a^{2}b^{4} to get 0.

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Factor

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