Question

question 8 of 10
multiply the following complex numbers:
(5 + 3i)(4 + 2i)
a. 26 - 22i
b. 14 + 22i
c. 26 + 22i
d. 14 - 22i

Explanation:

Step1: Use the distributive property (FOIL method)

To multiply two complex numbers \((a + bi)(c + di)\), we use the distributive property (FOIL):
\[

$$\begin{align*} (5 + 3i)(4 + 2i)&=5\times4 + 5\times2i + 3i\times4 + 3i\times2i\\ &=20 + 10i + 12i + 6i^2 \end{align*}$$

\]

Step2: Simplify using \(i^2 = -1\)

Recall that \(i^2 = -1\), so we substitute \(i^2\) with \(-1\) in the expression:
\[

$$\begin{align*} 20 + 10i + 12i + 6i^2&=20 + 10i + 12i + 6\times(-1)\\ &=20 + 10i + 12i - 6 \end{align*}$$

\]

Step3: Combine like terms

Combine the real parts and the imaginary parts separately:
\[

$$\begin{align*} (20 - 6) + (10i + 12i)&=14 + 22i \end{align*}$$

\]

Answer:

B. \(14 + 22i\)