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------------------------------------------------ Multiply and simplify the product: [tex](8-5i)^2[/tex].

Select the product:

A. 39
B. 89
C. 39 - [tex]80i[/tex]
D. [tex]89 - 80i[/tex]

Answer :

To solve the problem of multiplying and simplifying the expression [tex]\((8 - 5i)^2\)[/tex], we can use the formula for the square of a binomial. For a complex number in the form [tex]\((a + bi)\)[/tex], the square is given by:

[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]

Here, the real part [tex]\( a = 8 \)[/tex] and the imaginary part [tex]\( b = -5 \)[/tex].

Now, let's go step-by-step:

1. Calculate [tex]\( a^2 \)[/tex] (the square of the real part):
[tex]\[
a^2 = 8^2 = 64
\][/tex]

2. Calculate [tex]\( 2ab \)[/tex] (the middle term involving both real and imaginary parts):
[tex]\[
2ab = 2 \cdot 8 \cdot (-5) = 2 \cdot -40 = -80
\][/tex]

3. Calculate [tex]\((bi)^2\)[/tex] (the square of the imaginary part):
[tex]\[
(bi)^2 = (-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot (-1) = -25
\][/tex]
Remember that [tex]\(i^2 = -1\)[/tex].

4. Combine the results to form the expression [tex]\((8 - 5i)^2\)[/tex]:
- Add the real parts: [tex]\(64 + (-25) = 39\)[/tex]
- The imaginary part is [tex]\(-80i\)[/tex].

Thus, the product is:
[tex]\[
39 - 80i
\][/tex]

So, the correct answer is [tex]\(89 - 80i\)[/tex].