Solve 8(a^{2}-a)^3+42(a^2-a)^2-36a^2+36a

$$8 ( a ^ { 2 } - a ) ^ { 3 } + 42 ( a ^ { 2 } - a ) ^ { 2 } - 36 a ^ { 2 } + 36 a$$

Evaluate

$$2a\left(a-1\right)\left(2a-3\right)\left(2a+1\right)\left(a^{2}-a+6\right)$$

Solution Steps

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

$$8\left(\left(a^{2}\right)^{3}-3\left(a^{2}\right)^{2}a+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

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

$$8\left(a^{6}-3\left(a^{2}\right)^{2}a+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

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

$$8\left(a^{6}-3a^{4}a+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$8\left(a^{6}-3a^{5}+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$8\left(a^{6}-3a^{5}+3a^{4}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

Use the distributive property to multiply 8 by a^{6}-3a^{5}+3a^{4}-a^{3}.

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

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

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(\left(a^{2}\right)^{2}-2a^{2}a+a^{2}\right)-36a^{2}+36a$$

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

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(a^{4}-2a^{2}a+a^{2}\right)-36a^{2}+36a$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(a^{4}-2a^{3}+a^{2}\right)-36a^{2}+36a$$

Use the distributive property to multiply 42 by a^{4}-2a^{3}+a^{2}.

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42a^{4}-84a^{3}+42a^{2}-36a^{2}+36a$$

Combine 24a^{4} and 42a^{4} to get 66a^{4}.

$$8a^{6}-24a^{5}+66a^{4}-8a^{3}-84a^{3}+42a^{2}-36a^{2}+36a$$

Combine -8a^{3} and -84a^{3} to get -92a^{3}.

$$8a^{6}-24a^{5}+66a^{4}-92a^{3}+42a^{2}-36a^{2}+36a$$

Combine 42a^{2} and -36a^{2} to get 6a^{2}.

$$8a^{6}-24a^{5}+66a^{4}-92a^{3}+6a^{2}+36a$$

Expand

$$8a^{6}-24a^{5}+66a^{4}-92a^{3}+6a^{2}+36a$$

Solution Steps

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

$$8\left(\left(a^{2}\right)^{3}-3\left(a^{2}\right)^{2}a+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

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

$$8\left(a^{6}-3\left(a^{2}\right)^{2}a+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

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

$$8\left(a^{6}-3a^{4}a+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$8\left(a^{6}-3a^{5}+3a^{2}a^{2}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$8\left(a^{6}-3a^{5}+3a^{4}-a^{3}\right)+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

Use the distributive property to multiply 8 by a^{6}-3a^{5}+3a^{4}-a^{3}.

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(a^{2}-a\right)^{2}-36a^{2}+36a$$

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

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(\left(a^{2}\right)^{2}-2a^{2}a+a^{2}\right)-36a^{2}+36a$$

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

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(a^{4}-2a^{2}a+a^{2}\right)-36a^{2}+36a$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42\left(a^{4}-2a^{3}+a^{2}\right)-36a^{2}+36a$$

Use the distributive property to multiply 42 by a^{4}-2a^{3}+a^{2}.

$$8a^{6}-24a^{5}+24a^{4}-8a^{3}+42a^{4}-84a^{3}+42a^{2}-36a^{2}+36a$$

Combine 24a^{4} and 42a^{4} to get 66a^{4}.

$$8a^{6}-24a^{5}+66a^{4}-8a^{3}-84a^{3}+42a^{2}-36a^{2}+36a$$

Combine -8a^{3} and -84a^{3} to get -92a^{3}.

$$8a^{6}-24a^{5}+66a^{4}-92a^{3}+42a^{2}-36a^{2}+36a$$

Combine 42a^{2} and -36a^{2} to get 6a^{2}.

$$8a^{6}-24a^{5}+66a^{4}-92a^{3}+6a^{2}+36a$$