Question

lesson: multiplying complex numbers
simplify:
(3 + 5i) (2 − i)
○ 1 − 7i
○ 1 + 7i
○ 11 + 13i
● 11 + 7i

Explanation:

Step1: Apply distributive property (FOIL)

Multiply each term in the first complex number by each term in the second:
$(3 + 5i)(2 - i) = 3 \cdot 2 + 3 \cdot (-i) + 5i \cdot 2 + 5i \cdot (-i)$

Step2: Simplify each product

Calculate each term:
$3 \cdot 2 = 6$,
$3 \cdot (-i) = -3i$,
$5i \cdot 2 = 10i$,
$5i \cdot (-i) = -5i^2$.

Step3: Replace \(i^2\) with \(-1\)

Since \(i^2 = -1\), \(-5i^2 = -5(-1) = 5\).

Step4: Combine like terms

Combine real parts and imaginary parts:
Real: \(6 + 5 = 11\),
Imaginary: \(-3i + 10i = 7i\).

Step5: Form the simplified complex number

Combine real and imaginary parts: \(11 + 7i\).

Answer:

\(11 + 7i\) (corresponding to the option with the blue dot: \(11 + 7i\))