Solve a^2left(4a-3right)^2-aleft(4a-3right)^3

$${ a }^{ 2 } { \left(4a-3 \right) }^{ 2 } -a { \left( 4a-3 \right) }^{ 3 }$$

Evaluate

$$3a\left(1-a\right)\left(4a-3\right)^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-3\right)^{2}.

$$a^{2}\left(16a^{2}-24a+9\right)-a\left(4a-3\right)^{3}$$

Use the distributive property to multiply a^{2} by 16a^{2}-24a+9.

$$16a^{4}-24a^{3}+9a^{2}-a\left(4a-3\right)^{3}$$

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(4a-3\right)^{3}.

$$16a^{4}-24a^{3}+9a^{2}-a\left(64a^{3}-144a^{2}+108a-27\right)$$

Use the distributive property to multiply a by 64a^{3}-144a^{2}+108a-27.

$$16a^{4}-24a^{3}+9a^{2}-\left(64a^{4}-144a^{3}+108a^{2}-27a\right)$$

To find the opposite of 64a^{4}-144a^{3}+108a^{2}-27a, find the opposite of each term.

$$16a^{4}-24a^{3}+9a^{2}-64a^{4}+144a^{3}-108a^{2}+27a$$

Combine 16a^{4} and -64a^{4} to get -48a^{4}.

$$-48a^{4}-24a^{3}+9a^{2}+144a^{3}-108a^{2}+27a$$

Combine -24a^{3} and 144a^{3} to get 120a^{3}.

$$-48a^{4}+120a^{3}+9a^{2}-108a^{2}+27a$$

Combine 9a^{2} and -108a^{2} to get -99a^{2}.

$$-48a^{4}+120a^{3}-99a^{2}+27a$$

Expand

$$27a-99a^{2}+120a^{3}-48a^{4}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-3\right)^{2}.

$$a^{2}\left(16a^{2}-24a+9\right)-a\left(4a-3\right)^{3}$$

Use the distributive property to multiply a^{2} by 16a^{2}-24a+9.

$$16a^{4}-24a^{3}+9a^{2}-a\left(4a-3\right)^{3}$$

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(4a-3\right)^{3}.

$$16a^{4}-24a^{3}+9a^{2}-a\left(64a^{3}-144a^{2}+108a-27\right)$$

Use the distributive property to multiply a by 64a^{3}-144a^{2}+108a-27.

$$16a^{4}-24a^{3}+9a^{2}-\left(64a^{4}-144a^{3}+108a^{2}-27a\right)$$

To find the opposite of 64a^{4}-144a^{3}+108a^{2}-27a, find the opposite of each term.

$$16a^{4}-24a^{3}+9a^{2}-64a^{4}+144a^{3}-108a^{2}+27a$$

Combine 16a^{4} and -64a^{4} to get -48a^{4}.

$$-48a^{4}-24a^{3}+9a^{2}+144a^{3}-108a^{2}+27a$$

Combine -24a^{3} and 144a^{3} to get 120a^{3}.

$$-48a^{4}+120a^{3}+9a^{2}-108a^{2}+27a$$

Combine 9a^{2} and -108a^{2} to get -99a^{2}.

$$-48a^{4}+120a^{3}-99a^{2}+27a$$