To multiply the complex numbers [tex]\((4+i)\)[/tex] and [tex]\((4-i)\)[/tex], we can use the distributive property (also known as the FOIL method in this case, because we’re dealing with binomials).
Let’s write the numbers in standard form:
[tex]\[
(4+i)(4-i)
\][/tex]
Now apply the distributive property (FOIL method):
[tex]\[
(4+i)(4-i) = 4 \cdot 4 + 4 \cdot (-i) + i \cdot 4 + i \cdot (-i)
\][/tex]
Calculate each term separately:
1. [tex]\(4 \cdot 4 = 16\)[/tex]
2. [tex]\(4 \cdot (-i) = -4i\)[/tex]
3. [tex]\(i \cdot 4 = 4i\)[/tex]
4. [tex]\(i \cdot (-i) = -i^2\)[/tex]
Now, combine these results:
[tex]\[
(4+i)(4-i) = 16 - 4i + 4i - i^2
\][/tex]
Notice that [tex]\(-4i\)[/tex] and [tex]\(4i\)[/tex] cancel each other out:
[tex]\[
16 - i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
-i^2 = -(-1) = 1
\][/tex]
So the expression simplifies to:
[tex]\[
16 + 1 = 17
\][/tex]
Thus, the product of the complex numbers [tex]\((4+i)\)[/tex] and [tex]\((4-i)\)[/tex] is:
[tex]\[
17
\][/tex]
Therefore, the correct answer is:
[tex]\[
(D) 17
\][/tex]
