Solve 2(6k+2)^3-3(6k+2)^2+6k+2

$$2 ( 6 k + 2 ) ^ { 3 } - 3 ( 6 k + 2 ) ^ { 2 } + 6 k + 2$$

Evaluate

$$432k^{3}+324k^{2}+78k+6$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(6k+2\right)^{3}.

$$2\left(216k^{3}+216k^{2}+72k+8\right)-3\left(6k+2\right)^{2}+6k+2$$

Use the distributive property to multiply 2 by 216k^{3}+216k^{2}+72k+8.

$$432k^{3}+432k^{2}+144k+16-3\left(6k+2\right)^{2}+6k+2$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6k+2\right)^{2}.

$$432k^{3}+432k^{2}+144k+16-3\left(36k^{2}+24k+4\right)+6k+2$$

Use the distributive property to multiply -3 by 36k^{2}+24k+4.

$$432k^{3}+432k^{2}+144k+16-108k^{2}-72k-12+6k+2$$

Combine 432k^{2} and -108k^{2} to get 324k^{2}.

$$432k^{3}+324k^{2}+144k+16-72k-12+6k+2$$

Combine 144k and -72k to get 72k.

$$432k^{3}+324k^{2}+72k+16-12+6k+2$$

Subtract 12 from 16 to get 4.

$$432k^{3}+324k^{2}+72k+4+6k+2$$

Combine 72k and 6k to get 78k.

$$432k^{3}+324k^{2}+78k+4+2$$

Add 4 and 2 to get 6.

$$432k^{3}+324k^{2}+78k+6$$

Expand

$$432k^{3}+324k^{2}+78k+6$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(6k+2\right)^{3}.

$$2\left(216k^{3}+216k^{2}+72k+8\right)-3\left(6k+2\right)^{2}+6k+2$$

Use the distributive property to multiply 2 by 216k^{3}+216k^{2}+72k+8.

$$432k^{3}+432k^{2}+144k+16-3\left(6k+2\right)^{2}+6k+2$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6k+2\right)^{2}.

$$432k^{3}+432k^{2}+144k+16-3\left(36k^{2}+24k+4\right)+6k+2$$

Use the distributive property to multiply -3 by 36k^{2}+24k+4.

$$432k^{3}+432k^{2}+144k+16-108k^{2}-72k-12+6k+2$$

Combine 432k^{2} and -108k^{2} to get 324k^{2}.

$$432k^{3}+324k^{2}+144k+16-72k-12+6k+2$$

Combine 144k and -72k to get 72k.

$$432k^{3}+324k^{2}+72k+16-12+6k+2$$

Subtract 12 from 16 to get 4.

$$432k^{3}+324k^{2}+72k+4+6k+2$$

Combine 72k and 6k to get 78k.

$$432k^{3}+324k^{2}+78k+4+2$$

Add 4 and 2 to get 6.

$$432k^{3}+324k^{2}+78k+6$$