Two questions help on

Question

1. Multiply and write your answer as a complex number in simplest form (no spaces): (3 - 5i)(4 + 2i)

1. Multiply and write your answer as a complex number in simplest form (no spaces): (-4)(-8 + 5i)

Answer 1

(3 - 5i)(4 + 2i)

I use the FOIL approach here where FOIL stands for

F- first term [ in this case it is 3 and 4) - take the first terms, multiply them together

O- outside terms ( in this case it is 3 and 2i) - take the outside terms, multiply together and add to "F"

I - inside terms ( -5i and 4), take the inside terms, multiply together and add to "O"

L - last terms ( -5i and 2i), take the last terms, multiply together and add to "I"

Lastly combine and simplify for your final answer

(3 - 5i)(4 + 2i) = (3)(4) + (3)(2i) + (-5i)(4) + (-5i)(2i)

F O I L

where i*i = -1

= 12 + 6i -20i +(-5*2)(i*i)

= 12-14i+10

=22-14i ANS

(-4)(-8 + 5i)

Use distribution rule

(-4)(-8 + 5i) = (-4)(-8)+(-4)(5i)

= 32 -20i ANS

numbers or i, which is imaginerary with real numbers (a and b)

mean simplified or most reduced form

Answer 2

How to multiply complex numbers:

A complex number has the general form a + bi, where i is defined as √-1. The first thing to do is to understand the key properties of i--for example:

i*i = i² = -1

i³ = -i

i⁴ = +1

etc....

1/i = -i (prove this to yourself by replacing the 1 with -i² , and then dividing that by i.

When multiplying complex numbers, use the same rules as when multiplying expressions involving unknowns --eg (x + 2)*(x - 3). The big difference is that two "i terms" multiplied yields a real number.

Some examples:

(1 + i)(3- i) = 3 + 3i - i - (-1) = 4 + 2i

(5 - 2i)(6 +2i) = 30 - 12i + 10i -4(-1) = 33 + 8

To help understand all this, use an online calculator such as this one:

https://www.symbolab.com/solver/complex-numbers-calculator/%5Cleft(3%2B2i%5Cright)%5Cleft(3-4i%5Cright)

Answer 3

Use FOIL (First, Outer, Inner, Last)

(3 - 5i)(4 + 2i)

3(4) + 3(2i) - 5i(4) - 5i(2i)

12 + 6i - 20i - 10i²

12 - 14i - 10i² i² = -1

12 - 14i - 10(-1)

12 -14i + 10

22 - 14i

Answer 4

(3-5i)(4+2i)=12+6i-20i-10i^2

=12-14i-10i^2

or 10i^2-14i+12

Answer 5

For problem 1, you can use the FOIL method or you can use a generic rectangle. I'll use the rectangle:

So, combining the 4 portions of the rectangle, we get 12 - 20i + 6i - 10i² or 12 - 14i - 10i². Using the fact that i² = -1, we can simplify this to be: 12 - 14i - 10(-1) = 12 - 14i + 10 = 22-14i

For problem 2, just distribute the "-4" to the two elements inside the parenthesis like this:

(-4)(-8 + 5i)

(-4)(-8) + (-4)(5i)

32-20i