Question

what is the value of (7 + 2i)(3 - i)?
a. 19 - i
b. 21 - 2i
c. 21 + i
d. 23 - i

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:
$$(7 + 2i)(3 - i)=7\times3+7\times(-i)+2i\times3+2i\times(-i)$$

Step2: Simplify each product

Calculate each term:

• \(7\times3 = 21\)
• \(7\times(-i)=-7i\)
• \(2i\times3 = 6i\)
• \(2i\times(-i)=-2i^{2}\)

Recall that \(i^{2}=-1\), so \(-2i^{2}=-2\times(-1) = 2\)

Step3: Combine like terms

Combine the real parts and the imaginary parts:
Real parts: \(21 + 2=23\)
Imaginary parts: \(-7i+6i=-i\)
So, \((7 + 2i)(3 - i)=23 - i\)

Answer:

D. \(23 - i\)