Solve (x-1)*(x+1)*(x-1+i)*(x-1-i)

$$( x - 1 ) \cdot ( x + 1 ) \cdot ( x - 1 + i ) \cdot ( x - 1 - i )$$

Evaluate

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

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x+1.

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

Combine x and -x to get 0.

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

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

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

Combine -x^{2} and ix^{2} to get \left(-1+i\right)x^{2}.

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

Apply the distributive property by multiplying each term of x^{3}+\left(-1+i\right)x^{2}-x+1-i by each term of x-1-i.

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

Combine the real and imaginary parts.

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

Do the additions.

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

Combine -x^{3} and -ix^{3} to get \left(-1-i\right)x^{3}.

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

Combine \left(-1-i\right)x^{3} and \left(-1+i\right)x^{3} to get -2x^{3}.

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

Combine \left(1-i\right)x^{2} and \left(1+i\right)x^{2} to get 2x^{2}.

$$x^{4}-2x^{3}+2x^{2}-x^{2}+x+ix+x-ix-2$$

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

$$x^{4}-2x^{3}+x^{2}+x+ix+x-ix-2$$

Combine x and ix to get \left(1+i\right)x.

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

Combine \left(1+i\right)x and x to get \left(2+i\right)x.

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

Combine \left(2+i\right)x and -ix to get 2x.

$$x^{4}-2x^{3}+x^{2}+2x-2$$

Expand

$$x^{4}-2x^{3}+x^{2}+2x-2$$

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x+1.

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

Combine x and -x to get 0.

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

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

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

Combine -x^{2} and ix^{2} to get \left(-1+i\right)x^{2}.

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

Apply the distributive property by multiplying each term of x^{3}+\left(-1+i\right)x^{2}-x+1-i by each term of x-1-i.

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

Combine the real and imaginary parts.

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

Do the additions.

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

Combine -x^{3} and -ix^{3} to get \left(-1-i\right)x^{3}.

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

Combine \left(-1-i\right)x^{3} and \left(-1+i\right)x^{3} to get -2x^{3}.

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

Combine \left(1-i\right)x^{2} and \left(1+i\right)x^{2} to get 2x^{2}.

$$x^{4}-2x^{3}+2x^{2}-x^{2}+x+ix+x-ix-2$$

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

$$x^{4}-2x^{3}+x^{2}+x+ix+x-ix-2$$

Combine x and ix to get \left(1+i\right)x.

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

Combine \left(1+i\right)x and x to get \left(2+i\right)x.

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

Combine \left(2+i\right)x and -ix to get 2x.

$$x^{4}-2x^{3}+x^{2}+2x-2$$