Solve left(x-1right)left(x-1+4iright)left(x-1-4iright)

$$\left( x-1 \right) \left( x-1+4i \right) \left( x-1-4i \right)$$

Evaluate

$$\left(x-1\right)\left(x+\left(-1-4i\right)\right)\left(x+\left(-1+4i\right)\right)$$

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x-1+4i.

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

Combine -x and 4ix to get \left(-1+4i\right)x.

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

Combine \left(-1+4i\right)x and -x to get \left(-2+4i\right)x.

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

Apply the distributive property by multiplying each term of x^{2}+\left(-2+4i\right)x+1-4i by each term of x-1-4i.

$$x^{3}-x^{2}-4ix^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-1-4i-4ix+4i-16$$

Combine the real and imaginary parts.

$$x^{3}-x^{2}-4ix^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-1-16+\left(-4+4\right)i$$

Do the additions.

$$x^{3}-x^{2}-4ix^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-17$$

Combine -x^{2} and -4ix^{2} to get \left(-1-4i\right)x^{2}.

$$x^{3}+\left(-1-4i\right)x^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-17$$

Combine \left(-1-4i\right)x^{2} and \left(-2+4i\right)x^{2} to get -3x^{2}.

$$x^{3}-3x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-17$$

Combine \left(2-4i\right)x and \left(16+8i\right)x to get \left(18+4i\right)x.

$$x^{3}-3x^{2}+\left(18+4i\right)x+x-4ix-17$$

Combine \left(18+4i\right)x and x to get \left(19+4i\right)x.

$$x^{3}-3x^{2}+\left(19+4i\right)x-4ix-17$$

Combine \left(19+4i\right)x and -4ix to get 19x.

$$x^{3}-3x^{2}+19x-17$$

Expand

$$x^{3}-3x^{2}+19x-17$$

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x-1+4i.

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

Combine -x and 4ix to get \left(-1+4i\right)x.

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

Combine \left(-1+4i\right)x and -x to get \left(-2+4i\right)x.

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

Apply the distributive property by multiplying each term of x^{2}+\left(-2+4i\right)x+1-4i by each term of x-1-4i.

$$x^{3}-x^{2}-4ix^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-1-4i-4ix+4i-16$$

Combine the real and imaginary parts.

$$x^{3}-x^{2}-4ix^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-1-16+\left(-4+4\right)i$$

Do the additions.

$$x^{3}-x^{2}-4ix^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-17$$

Combine -x^{2} and -4ix^{2} to get \left(-1-4i\right)x^{2}.

$$x^{3}+\left(-1-4i\right)x^{2}+\left(-2+4i\right)x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-17$$

Combine \left(-1-4i\right)x^{2} and \left(-2+4i\right)x^{2} to get -3x^{2}.

$$x^{3}-3x^{2}+\left(2-4i\right)x+\left(16+8i\right)x+x-4ix-17$$

Combine \left(2-4i\right)x and \left(16+8i\right)x to get \left(18+4i\right)x.

$$x^{3}-3x^{2}+\left(18+4i\right)x+x-4ix-17$$

Combine \left(18+4i\right)x and x to get \left(19+4i\right)x.

$$x^{3}-3x^{2}+\left(19+4i\right)x-4ix-17$$

Combine \left(19+4i\right)x and -4ix to get 19x.

$$x^{3}-3x^{2}+19x-17$$