Solve (x-3)^3-(x+3)(x^2-3x+9)

$$( x - 3 ) ^ { 3 } - ( x + 3 ) ( x ^ { 2 } - 3 x + 9 )$$

Evaluate

$$-9x^{2}+27x-54$$

Solution Steps

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

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

Use the distributive property to multiply x+3 by x^{2}-3x+9 and combine like terms.

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

To find the opposite of x^{3}+27, find the opposite of each term.

$$x^{3}-9x^{2}+27x-27-x^{3}-27$$

Combine x^{3} and -x^{3} to get 0.

$$-9x^{2}+27x-27-27$$

Subtract 27 from -27 to get -54.

$$-9x^{2}+27x-54$$

Expand

$$-9x^{2}+27x-54$$

Solution Steps

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

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

Use the distributive property to multiply x+3 by x^{2}-3x+9 and combine like terms.

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

To find the opposite of x^{3}+27, find the opposite of each term.

$$x^{3}-9x^{2}+27x-27-x^{3}-27$$

Combine x^{3} and -x^{3} to get 0.

$$-9x^{2}+27x-27-27$$

Subtract 27 from -27 to get -54.

$$-9x^{2}+27x-54$$