Solve (f+g)=2(3x-1)^3+2(3x-1)

$$( f + g ) = 2 ( 3 x - 1 ) ^ { 3 } + 2 ( 3 x - 1 )$$

Solve for f

$$f=-\left(-2\left(3x-1\right)^{3}+g-6x+2\right)$$

Solution Steps

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

$$f+g=2\left(27x^{3}-27x^{2}+9x-1\right)+2\left(3x-1\right)$$

Use the distributive property to multiply 2 by 27x^{3}-27x^{2}+9x-1.

$$f+g=54x^{3}-54x^{2}+18x-2+2\left(3x-1\right)$$

Use the distributive property to multiply 2 by 3x-1.

$$f+g=54x^{3}-54x^{2}+18x-2+6x-2$$

Combine 18x and 6x to get 24x.

$$f+g=54x^{3}-54x^{2}+24x-2-2$$

Subtract 2 from -2 to get -4.

$$f+g=54x^{3}-54x^{2}+24x-4$$

Subtract g from both sides.

$$f=54x^{3}-54x^{2}+24x-4-g$$

Solve for g

$$g=-\left(-2\left(3x-1\right)^{3}+f-6x+2\right)$$

Solution Steps

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

$$f+g=2\left(27x^{3}-27x^{2}+9x-1\right)+2\left(3x-1\right)$$

Use the distributive property to multiply 2 by 27x^{3}-27x^{2}+9x-1.

$$f+g=54x^{3}-54x^{2}+18x-2+2\left(3x-1\right)$$

Use the distributive property to multiply 2 by 3x-1.

$$f+g=54x^{3}-54x^{2}+18x-2+6x-2$$

Combine 18x and 6x to get 24x.

$$f+g=54x^{3}-54x^{2}+24x-2-2$$

Subtract 2 from -2 to get -4.

$$f+g=54x^{3}-54x^{2}+24x-4$$

Subtract f from both sides.

$$g=54x^{3}-54x^{2}+24x-4-f$$