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

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

Expand

$$x^{3}-7x^{2}+14x-9$$

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}.

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

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

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

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

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

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$x^{3}-7x^{2}+12x-8+2x-1$$

Combine 12x and 2x to get 14x.

$$x^{3}-7x^{2}+14x-8-1$$

Subtract 1 from -8 to get -9.

$$x^{3}-7x^{2}+14x-9$$

Evaluate

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

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}.

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

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

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

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

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

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$x^{3}-7x^{2}+12x-8+2x-1$$

Combine 12x and 2x to get 14x.

$$x^{3}-7x^{2}+14x-8-1$$

Subtract 1 from -8 to get -9.

$$x^{3}-7x^{2}+14x-9$$