Let's break down this problem step by step. The task is to find the product of two complex numbers, [tex]\( z_1 = 5 + 8i \)[/tex] and [tex]\( z_2 = -9 - 2i \)[/tex].
To find the product of two complex numbers, we use the distributive property, often represented by the FOIL method in this context, where each part of the first complex number is multiplied by each part of the second complex number.
Given:
[tex]\[ z_1 = 5 + 8i \][/tex]
[tex]\[ z_2 = -9 - 2i \][/tex]
Now, compute their product:
[tex]\[ (5 + 8i)(-9 - 2i) \][/tex]
First, we multiply the real and imaginary parts separately:
[tex]\[ 5 \times (-9) = -45 \][/tex]
[tex]\[ 5 \times (-2i) = -10i \][/tex]
[tex]\[ 8i \times (-9) = -72i \][/tex]
[tex]\[ 8i \times (-2i) = -16i^2 \][/tex]
Recall that [tex]\( i^2 = -1 \)[/tex].
So,
[tex]\[ -16i^2 = -16 \times (-1) = 16 \][/tex]
Now, sum up all the parts we computed:
[tex]\[ -45 + (-10i) + (-72i) + 16 \][/tex]
Combine the real parts and the imaginary parts:
[tex]\[ (-45 + 16) + (-10i - 72i) \][/tex]
Simplify:
[tex]\[ -29 + (-82i) \][/tex]
Thus, the product of [tex]\( 5 + 8i \)[/tex] and [tex]\( -9 - 2i \)[/tex] is:
[tex]\[ -29 - 82i \][/tex]
So the real part of the product is [tex]\(-29\)[/tex] and the imaginary part is [tex]\(-82\)[/tex].
