Question

multiply.
(1 - 2i)(5 - i)
write your answer as a complex number in standard form.

Explanation:

Step1: Apply the distributive property (FOIL method)

We multiply each term in the first complex number by each term in the second complex number:
$$(1 - 2i)(5 - i)=1\times5+1\times(-i)+(-2i)\times5+(-2i)\times(-i)$$

Step2: Simplify each product

Simplify each of the four products:

• \(1\times5 = 5\)
• \(1\times(-i)=-i\)
• \((-2i)\times5=-10i\)
• \((-2i)\times(-i) = 2i^{2}\) (since multiplying two negative numbers gives a positive, and \(i\times i = i^{2}\))

So now we have:
$$5 - i-10i + 2i^{2}$$

Step3: Recall that \(i^{2}=-1\)

Substitute \(i^{2}=-1\) into the expression:
$$5 - i-10i+2\times(-1)$$

Step4: Simplify the real and imaginary parts

First, simplify the real parts: \(5 + 2\times(-1)=5 - 2 = 3\)
Then, simplify the imaginary parts: \(-i-10i=-11i\)
Combine the real and imaginary parts:
$$3-11i$$

Answer:

\(3 - 11i\)