Question

What is the value of left-parenthesis 7 minus 3 i right-parenthesis left-parenthesis 6 plus i right-parenthesis?

Answer options with 4 options
A.
42 minus 14 i

B.
45 minus 11 i

C.
45 minus 25 i

D.
42 minus 28 i

Answer 1

To find the value of \((7 - 3i)(6 + i)\), we can use the distributive property (also known as the FOIL method for binomials).

Let's calculate it step by step:

1. **First**: Multiply the first terms:
\[
7 \cdot 6 = 42
\]

2. **Outer**: Multiply the outer terms:
\[
7 \cdot i = 7i
\]

3. **Inner**: Multiply the inner terms:
\[
-3i \cdot 6 = -18i
\]

4. **Last**: Multiply the last terms:
\[
-3i \cdot i = -3i^2
\]
Since \(i^2 = -1\), we have:
\[
-3i^2 = -3(-1) = 3
\]

Now, we can combine all these results:
\[
(7 - 3i)(6 + i) = 42 + 7i - 18i + 3
\]

Combining like terms:
- Real part: \(42 + 3 = 45\)
- Imaginary part: \(7i - 18i = -11i\)

So, we have:
\[
(7 - 3i)(6 + i) = 45 - 11i
\]

Thus, the answer is **B. \(45 - 11i\)**.