Solve (2a+1)^3-(a+1)^3-3a(3a+1)

$$( 2 a + 1 ) ^ { 3 } - ( a + 1 ) ^ { 3 } - 3 a ( 3 a + 1 )$$

Evaluate

$$7a^{3}$$

Solution Steps

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

$$8a^{3}+12a^{2}+6a+1-\left(a+1\right)^{3}-3a\left(3a+1\right)$$

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

$$8a^{3}+12a^{2}+6a+1-\left(a^{3}+3a^{2}+3a+1\right)-3a\left(3a+1\right)$$

To find the opposite of a^{3}+3a^{2}+3a+1, find the opposite of each term.

$$8a^{3}+12a^{2}+6a+1-a^{3}-3a^{2}-3a-1-3a\left(3a+1\right)$$

Combine 8a^{3} and -a^{3} to get 7a^{3}.

$$7a^{3}+12a^{2}+6a+1-3a^{2}-3a-1-3a\left(3a+1\right)$$

Combine 12a^{2} and -3a^{2} to get 9a^{2}.

$$7a^{3}+9a^{2}+6a+1-3a-1-3a\left(3a+1\right)$$

Combine 6a and -3a to get 3a.

$$7a^{3}+9a^{2}+3a+1-1-3a\left(3a+1\right)$$

Subtract 1 from 1 to get 0.

$$7a^{3}+9a^{2}+3a-3a\left(3a+1\right)$$

Use the distributive property to multiply -3a by 3a+1.

$$7a^{3}+9a^{2}+3a-9a^{2}-3a$$

Combine 9a^{2} and -9a^{2} to get 0.

$$7a^{3}+3a-3a$$

Combine 3a and -3a to get 0.

$$7a^{3}$$

Expand

$$7a^{3}$$

Solution Steps

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

$$8a^{3}+12a^{2}+6a+1-\left(a+1\right)^{3}-3a\left(3a+1\right)$$

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

$$8a^{3}+12a^{2}+6a+1-\left(a^{3}+3a^{2}+3a+1\right)-3a\left(3a+1\right)$$

To find the opposite of a^{3}+3a^{2}+3a+1, find the opposite of each term.

$$8a^{3}+12a^{2}+6a+1-a^{3}-3a^{2}-3a-1-3a\left(3a+1\right)$$

Combine 8a^{3} and -a^{3} to get 7a^{3}.

$$7a^{3}+12a^{2}+6a+1-3a^{2}-3a-1-3a\left(3a+1\right)$$

Combine 12a^{2} and -3a^{2} to get 9a^{2}.

$$7a^{3}+9a^{2}+6a+1-3a-1-3a\left(3a+1\right)$$

Combine 6a and -3a to get 3a.

$$7a^{3}+9a^{2}+3a+1-1-3a\left(3a+1\right)$$

Subtract 1 from 1 to get 0.

$$7a^{3}+9a^{2}+3a-3a\left(3a+1\right)$$

Use the distributive property to multiply -3a by 3a+1.

$$7a^{3}+9a^{2}+3a-9a^{2}-3a$$

Combine 9a^{2} and -9a^{2} to get 0.

$$7a^{3}+3a-3a$$

Combine 3a and -3a to get 0.

$$7a^{3}$$