Question

What is the product of left-parenthesis negative 7 plus 4 i right-parenthesis times left-parenthesis 1 minus 8 i right-parenthesis?

Answer options with 4 options
A.
negative 39 minus 52 i

B.
negative 39 plus 60 i

C.
25 minus 52 i

D.
25 plus 60 i

Answer 1

To find the product of \(( -7 + 4i )\) and \(( 1 - 8i )\), we can use the distributive property (also known as the FOIL method for binomials).

\[
(-7 + 4i)(1 - 8i)
\]

Calculating this step by step:

1. **First Term**:
\[
-7 \cdot 1 = -7
\]

2. **Outer Term**:
\[
-7 \cdot (-8i) = 56i
\]

3. **Inner Term**:
\[
4i \cdot 1 = 4i
\]

4. **Last Term**:
\[
4i \cdot (-8i) = -32i^2
\]

Since \(i^2 = -1\), we have:
\[
-32i^2 = -32(-1) = 32
\]

Now combine all the results:
\[
-7 + 56i + 4i + 32
\]

Combining the real parts and the imaginary parts:
- Real parts: \(-7 + 32 = 25\)
- Imaginary parts: \(56i + 4i = 60i\)

Putting it all together:
\[
25 + 60i
\]

Thus, the product is:
\[
\boxed{25 + 60i}
\]

The correct answer is **D. 25 plus 60 i**.