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

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

Evaluate

$$\left(7-2x\right)\left(x-2\right)^{2}$$

Solution Steps

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

$$x^{3}-6x^{2}+12x-8-\left(3x-9\right)\left(x-2\right)^{2}$$

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

$$x^{3}-6x^{2}+12x-8-\left(3x-9\right)\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply 3x-9 by x^{2}-4x+4 and combine like terms.

$$x^{3}-6x^{2}+12x-8-\left(3x^{3}-21x^{2}+48x-36\right)$$

To find the opposite of 3x^{3}-21x^{2}+48x-36, find the opposite of each term.

$$x^{3}-6x^{2}+12x-8-3x^{3}+21x^{2}-48x+36$$

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

$$-2x^{3}-6x^{2}+12x-8+21x^{2}-48x+36$$

Combine -6x^{2} and 21x^{2} to get 15x^{2}.

$$-2x^{3}+15x^{2}+12x-8-48x+36$$

Combine 12x and -48x to get -36x.

$$-2x^{3}+15x^{2}-36x-8+36$$

Add -8 and 36 to get 28.

$$-2x^{3}+15x^{2}-36x+28$$

Expand

$$28-36x+15x^{2}-2x^{3}$$

Solution Steps

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

$$x^{3}-6x^{2}+12x-8-\left(3x-9\right)\left(x-2\right)^{2}$$

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

$$x^{3}-6x^{2}+12x-8-\left(3x-9\right)\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply 3x-9 by x^{2}-4x+4 and combine like terms.

$$x^{3}-6x^{2}+12x-8-\left(3x^{3}-21x^{2}+48x-36\right)$$

To find the opposite of 3x^{3}-21x^{2}+48x-36, find the opposite of each term.

$$x^{3}-6x^{2}+12x-8-3x^{3}+21x^{2}-48x+36$$

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

$$-2x^{3}-6x^{2}+12x-8+21x^{2}-48x+36$$

Combine -6x^{2} and 21x^{2} to get 15x^{2}.

$$-2x^{3}+15x^{2}+12x-8-48x+36$$

Combine 12x and -48x to get -36x.

$$-2x^{3}+15x^{2}-36x-8+36$$

Add -8 and 36 to get 28.

$$-2x^{3}+15x^{2}-36x+28$$