91Ó°ÊÓ

To understand complex number multiplication, we need to start with the basics. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which satisfies \(i^2 = -1\).

When you multiply two complex numbers, you're essentially using the same principles as you would with binomials. For example, to multiply \((3+2i)\) and \((5-i)\), we need to employ the distributive property, also known as the FOIL (First, Outer, Inner, Last) method. This means that each term in the first complex number must be multiplied by each term in the second complex number.

By doing so, you'll get:
- First: \(3 \times 5 = 15\)
- Outer: \(3 \times -i = -3i\)
- Inner: \(2i \times 5 = 10i\)
- Last: \(2i \times -i = -2i^2\)

The distributive property is a fundamental arithmetic principle that is crucial when dealing with expressions involving complex numbers. The property states that for any three numbers \(a\), \(b\), and \(c\), we have \(a(b + c) = ab + ac\).

Applying this to complex numbers, let's revisit our example of multiplying \((3+2i)\) and \((5-i)\). We distribute both parts of the first complex number across both parts of the second complex number:

\((3+2i)(5-i) \) becomes \(3(5) + 3(-i) + 2i(5) + 2i(-i)\).

Each term is then multiplied independently:
- 3 times 5 gives \(15\)
- 3 times \(-i\) gives \(-3i\)
- 2i times 5 gives \(10i\)
- 2i times \(-i\) gives \(-2i^2\)

Crucial to solving complex number multiplication is the simplification of terms involving the imaginary unit \(i\). Remember, the imaginary unit has the property \(i^2 = -1\).

In our multiplication problem, after distributing terms, we end up with \(-2i^2\). Since \(i^2 = -1\), this simplifies to \(-2(-1)\), which is \( 2 \). Thus, the term \(-2i^2\) simplifies to just \(2\).

The real parts are then combined:
- \(15\) (from the real multiplication) plus \(2\) (from the simplified \(-2i^2\) term) gives \(17\)

The imaginary parts are combined as well:
- \(-3i + 10i\) gives \(7i\)

Put together, the result of \((3+2i)(5-i)\) is \(17 + 7i\). Here, the real part \(a\) is \(17\) which corresponds to option (D) in the problem.