Solve 3(3x+2)^3+2(3x+2)^2-3(3x+2)-2

$$3 ( 3 x + 2 ) ^ { 3 } + 2 ( 3 x + 2 ) ^ { 2 } - 3 ( 3 x + 2 ) - 2$$

Evaluate

$$3\left(x+1\right)\left(3x+1\right)\left(9x+8\right)$$

Solution Steps

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

$$3\left(27x^{3}+54x^{2}+36x+8\right)+2\left(3x+2\right)^{2}-3\left(3x+2\right)-2$$

Use the distributive property to multiply 3 by 27x^{3}+54x^{2}+36x+8.

$$81x^{3}+162x^{2}+108x+24+2\left(3x+2\right)^{2}-3\left(3x+2\right)-2$$

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

$$81x^{3}+162x^{2}+108x+24+2\left(9x^{2}+12x+4\right)-3\left(3x+2\right)-2$$

Use the distributive property to multiply 2 by 9x^{2}+12x+4.

$$81x^{3}+162x^{2}+108x+24+18x^{2}+24x+8-3\left(3x+2\right)-2$$

Combine 162x^{2} and 18x^{2} to get 180x^{2}.

$$81x^{3}+180x^{2}+108x+24+24x+8-3\left(3x+2\right)-2$$

Combine 108x and 24x to get 132x.

$$81x^{3}+180x^{2}+132x+24+8-3\left(3x+2\right)-2$$

Add 24 and 8 to get 32.

$$81x^{3}+180x^{2}+132x+32-3\left(3x+2\right)-2$$

Use the distributive property to multiply -3 by 3x+2.

$$81x^{3}+180x^{2}+132x+32-9x-6-2$$

Combine 132x and -9x to get 123x.

$$81x^{3}+180x^{2}+123x+32-6-2$$

Subtract 6 from 32 to get 26.

$$81x^{3}+180x^{2}+123x+26-2$$

Subtract 2 from 26 to get 24.

$$81x^{3}+180x^{2}+123x+24$$

Expand

$$81x^{3}+180x^{2}+123x+24$$

Solution Steps

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

$$3\left(27x^{3}+54x^{2}+36x+8\right)+2\left(3x+2\right)^{2}-3\left(3x+2\right)-2$$

Use the distributive property to multiply 3 by 27x^{3}+54x^{2}+36x+8.

$$81x^{3}+162x^{2}+108x+24+2\left(3x+2\right)^{2}-3\left(3x+2\right)-2$$

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

$$81x^{3}+162x^{2}+108x+24+2\left(9x^{2}+12x+4\right)-3\left(3x+2\right)-2$$

Use the distributive property to multiply 2 by 9x^{2}+12x+4.

$$81x^{3}+162x^{2}+108x+24+18x^{2}+24x+8-3\left(3x+2\right)-2$$

Combine 162x^{2} and 18x^{2} to get 180x^{2}.

$$81x^{3}+180x^{2}+108x+24+24x+8-3\left(3x+2\right)-2$$

Combine 108x and 24x to get 132x.

$$81x^{3}+180x^{2}+132x+24+8-3\left(3x+2\right)-2$$

Add 24 and 8 to get 32.

$$81x^{3}+180x^{2}+132x+32-3\left(3x+2\right)-2$$

Use the distributive property to multiply -3 by 3x+2.

$$81x^{3}+180x^{2}+132x+32-9x-6-2$$

Combine 132x and -9x to get 123x.

$$81x^{3}+180x^{2}+123x+32-6-2$$

Subtract 6 from 32 to get 26.

$$81x^{3}+180x^{2}+123x+26-2$$

Subtract 2 from 26 to get 24.

$$81x^{3}+180x^{2}+123x+24$$