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

$$( x - 1 ) ( x - 2 ) ( x - 2 ) ( x - 3 )$$

Evaluate

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

Solution Steps

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

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

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

$$\left(x-1\right)\left(x^{2}-4x+4\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x-1 by each term of x^{2}-4x+4.

$$\left(x^{3}-4x^{2}+4x-x^{2}+4x-4\right)\left(x-3\right)$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$\left(x^{3}-5x^{2}+4x+4x-4\right)\left(x-3\right)$$

Combine 4x and 4x to get 8x.

$$\left(x^{3}-5x^{2}+8x-4\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x^{3}-5x^{2}+8x-4 by each term of x-3.

$$x^{4}-3x^{3}-5x^{3}+15x^{2}+8x^{2}-24x-4x+12$$

Combine -3x^{3} and -5x^{3} to get -8x^{3}.

$$x^{4}-8x^{3}+15x^{2}+8x^{2}-24x-4x+12$$

Combine 15x^{2} and 8x^{2} to get 23x^{2}.

$$x^{4}-8x^{3}+23x^{2}-24x-4x+12$$

Combine -24x and -4x to get -28x.

$$x^{4}-8x^{3}+23x^{2}-28x+12$$

Expand

$$x^{4}-8x^{3}+23x^{2}-28x+12$$

Solution Steps

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

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

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

$$\left(x-1\right)\left(x^{2}-4x+4\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x-1 by each term of x^{2}-4x+4.

$$\left(x^{3}-4x^{2}+4x-x^{2}+4x-4\right)\left(x-3\right)$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$\left(x^{3}-5x^{2}+4x+4x-4\right)\left(x-3\right)$$

Combine 4x and 4x to get 8x.

$$\left(x^{3}-5x^{2}+8x-4\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x^{3}-5x^{2}+8x-4 by each term of x-3.

$$x^{4}-3x^{3}-5x^{3}+15x^{2}+8x^{2}-24x-4x+12$$

Combine -3x^{3} and -5x^{3} to get -8x^{3}.

$$x^{4}-8x^{3}+15x^{2}+8x^{2}-24x-4x+12$$

Combine 15x^{2} and 8x^{2} to get 23x^{2}.

$$x^{4}-8x^{3}+23x^{2}-24x-4x+12$$

Combine -24x and -4x to get -28x.

$$x^{4}-8x^{3}+23x^{2}-28x+12$$