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

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

Evaluate

$$9x^{3}-14x^{2}+\frac{19x}{3}-\frac{26}{27}$$

Solution Steps

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$\left(x+\frac{1}{3}\right)^{3}-3x^{2}-\left(1-2x\right)^{3}$$

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

$$x^{3}+x^{2}+\frac{1}{3}x+\frac{1}{27}-3x^{2}-\left(1-2x\right)^{3}$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$x^{3}-2x^{2}+\frac{1}{3}x+\frac{1}{27}-\left(1-2x\right)^{3}$$

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

$$x^{3}-2x^{2}+\frac{1}{3}x+\frac{1}{27}-\left(1-6x+12x^{2}-8x^{3}\right)$$

To find the opposite of 1-6x+12x^{2}-8x^{3}, find the opposite of each term.

$$x^{3}-2x^{2}+\frac{1}{3}x+\frac{1}{27}-1+6x-12x^{2}+8x^{3}$$

Subtract 1 from \frac{1}{27} to get -\frac{26}{27}.

$$x^{3}-2x^{2}+\frac{1}{3}x-\frac{26}{27}+6x-12x^{2}+8x^{3}$$

Combine \frac{1}{3}x and 6x to get \frac{19}{3}x.

$$x^{3}-2x^{2}+\frac{19}{3}x-\frac{26}{27}-12x^{2}+8x^{3}$$

Combine -2x^{2} and -12x^{2} to get -14x^{2}.

$$x^{3}-14x^{2}+\frac{19}{3}x-\frac{26}{27}+8x^{3}$$

Combine x^{3} and 8x^{3} to get 9x^{3}.

$$9x^{3}-14x^{2}+\frac{19}{3}x-\frac{26}{27}$$

Expand

$$9x^{3}-14x^{2}+\frac{19x}{3}-\frac{26}{27}$$

Solution Steps

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$\left(x+\frac{1}{3}\right)^{3}-3x^{2}-\left(1-2x\right)^{3}$$

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

$$x^{3}+x^{2}+\frac{1}{3}x+\frac{1}{27}-3x^{2}-\left(1-2x\right)^{3}$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$x^{3}-2x^{2}+\frac{1}{3}x+\frac{1}{27}-\left(1-2x\right)^{3}$$

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

$$x^{3}-2x^{2}+\frac{1}{3}x+\frac{1}{27}-\left(1-6x+12x^{2}-8x^{3}\right)$$

To find the opposite of 1-6x+12x^{2}-8x^{3}, find the opposite of each term.

$$x^{3}-2x^{2}+\frac{1}{3}x+\frac{1}{27}-1+6x-12x^{2}+8x^{3}$$

Subtract 1 from \frac{1}{27} to get -\frac{26}{27}.

$$x^{3}-2x^{2}+\frac{1}{3}x-\frac{26}{27}+6x-12x^{2}+8x^{3}$$

Combine \frac{1}{3}x and 6x to get \frac{19}{3}x.

$$x^{3}-2x^{2}+\frac{19}{3}x-\frac{26}{27}-12x^{2}+8x^{3}$$

Combine -2x^{2} and -12x^{2} to get -14x^{2}.

$$x^{3}-14x^{2}+\frac{19}{3}x-\frac{26}{27}+8x^{3}$$

Combine x^{3} and 8x^{3} to get 9x^{3}.

$$9x^{3}-14x^{2}+\frac{19}{3}x-\frac{26}{27}$$