To determine which operation results in the complex number [tex]\(10 - 5i\)[/tex] using the given complex numbers [tex]\(4 + 3i\)[/tex] and [tex]\(1 - 2i\)[/tex], we need to consider the multiplication of these complex numbers. Let's break down each step in detail:
1. Write down the complex numbers:
[tex]\[
z_1 = 4 + 3i
\][/tex]
[tex]\[
z_2 = 1 - 2i
\][/tex]
2. Multiply these two complex numbers:
To multiply two complex numbers, use the distributive property:
[tex]\[
z_1 \cdot z_2 = (4 + 3i)(1 - 2i)
\][/tex]
3. Distribute each term in the first complex number by each term in the second complex number:
[tex]\[
(4 + 3i)(1 - 2i) = 4 \cdot 1 + 4 \cdot (-2i) + 3i \cdot 1 + 3i \cdot (-2i)
\][/tex]
4. Calculate each term individually:
[tex]\[
= 4 \cdot 1 + 4 \cdot (-2i) + 3i \cdot 1 + 3i \cdot (-2i)
\][/tex]
[tex]\[
= 4 - 8i + 3i - 6i^2
\][/tex]
5. Combine the like terms and remember that [tex]\(i^2 = -1\)[/tex]:
Substitute [tex]\(-1\)[/tex] for [tex]\(i^2\)[/tex]:
[tex]\[
= 4 - 8i + 3i - 6(-1)
\][/tex]
6. Simplify the expression:
[tex]\[
= 4 - 8i + 3i + 6
\][/tex]
Combine the real parts and the imaginary parts:
[tex]\[
= (4 + 6) + (-8i + 3i)
\][/tex]
[tex]\[
= 10 - 5i
\][/tex]
Hence, the operation that results in the complex number [tex]\(10 - 5i\)[/tex] is the multiplication of [tex]\(4 + 3i\)[/tex] and [tex]\(1 - 2i\)[/tex].
