Solve 3p^2-2p^3+6p^2left(1-pright)^3+3{p}^3{left(1-pright)}^2

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

Evaluate

$$p^{2}\left(9-17p+12p^{2}-3p^{3}\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(1-p\right)^{3}.

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

Use the distributive property to multiply 6p^{2} by 1-3p+3p^{2}-p^{3}.

$$3p^{2}-2p^{3}+6p^{2}-18p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-p\right)^{2}$$

Combine 3p^{2} and 6p^{2} to get 9p^{2}.

$$9p^{2}-2p^{3}-18p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-p\right)^{2}$$

Combine -2p^{3} and -18p^{3} to get -20p^{3}.

$$9p^{2}-20p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-p\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-p\right)^{2}.

$$9p^{2}-20p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-2p+p^{2}\right)$$

Use the distributive property to multiply 3p^{3} by 1-2p+p^{2}.

$$9p^{2}-20p^{3}+18p^{4}-6p^{5}+3p^{3}-6p^{4}+3p^{5}$$

Combine -20p^{3} and 3p^{3} to get -17p^{3}.

$$9p^{2}-17p^{3}+18p^{4}-6p^{5}-6p^{4}+3p^{5}$$

Combine 18p^{4} and -6p^{4} to get 12p^{4}.

$$9p^{2}-17p^{3}+12p^{4}-6p^{5}+3p^{5}$$

Combine -6p^{5} and 3p^{5} to get -3p^{5}.

$$9p^{2}-17p^{3}+12p^{4}-3p^{5}$$

Expand

$$9p^{2}-17p^{3}+12p^{4}-3p^{5}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(1-p\right)^{3}.

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

Use the distributive property to multiply 6p^{2} by 1-3p+3p^{2}-p^{3}.

$$3p^{2}-2p^{3}+6p^{2}-18p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-p\right)^{2}$$

Combine 3p^{2} and 6p^{2} to get 9p^{2}.

$$9p^{2}-2p^{3}-18p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-p\right)^{2}$$

Combine -2p^{3} and -18p^{3} to get -20p^{3}.

$$9p^{2}-20p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-p\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-p\right)^{2}.

$$9p^{2}-20p^{3}+18p^{4}-6p^{5}+3p^{3}\left(1-2p+p^{2}\right)$$

Use the distributive property to multiply 3p^{3} by 1-2p+p^{2}.

$$9p^{2}-20p^{3}+18p^{4}-6p^{5}+3p^{3}-6p^{4}+3p^{5}$$

Combine -20p^{3} and 3p^{3} to get -17p^{3}.

$$9p^{2}-17p^{3}+18p^{4}-6p^{5}-6p^{4}+3p^{5}$$

Combine 18p^{4} and -6p^{4} to get 12p^{4}.

$$9p^{2}-17p^{3}+12p^{4}-6p^{5}+3p^{5}$$

Combine -6p^{5} and 3p^{5} to get -3p^{5}.

$$9p^{2}-17p^{3}+12p^{4}-3p^{5}$$