Question

What is the value of left-parenthesis 7 plus 2 i right-parenthesis left-parenthesis 3 minus i right-parenthesis?

Answer options with 4 options
A.
19 minus i

B.
21 minus 2 i

C.
21 plus i

D.
23 minus i

Answer 1

To find the value of \((7 + 2i)(3 - i)\), we can use the distributive property (also known as the FOIL method for binomials).

Let's calculate it step by step:

1. Multiply the first terms:
\[
7 \times 3 = 21
\]

2. Multiply the outer terms:
\[
7 \times (-i) = -7i
\]

3. Multiply the inner terms:
\[
2i \times 3 = 6i
\]

4. Multiply the last terms:
\[
2i \times (-i) = -2i^2
\]
Since \(i^2 = -1\), we have:
\[
-2i^2 = -2(-1) = 2
\]

Now we can combine all these results:
\[
21 - 7i + 6i + 2
\]

Combine the like terms:
- The real parts: \(21 + 2 = 23\)
- The imaginary parts: \(-7i + 6i = -i\)

Putting it all together, we get:
\[
23 - i
\]

Thus, the value of \((7 + 2i)(3 - i)\) is \(23 - i\).

Looking at the answer options provided:
A. \(19 - i\)
B. \(21 - 2i\)
C. \(21 + i\)
D. \(23 - i\)

The correct answer is:
**D. \(23 - i\)**