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Multiplying Complex Numbers

Multiplying complex numbers is very much like multiplying polynomials. When we simplify the result, we replace each occurrence of i² with -1. We often write the final result in the form a + bi.

Example 1

Find: 5i Â· 7i

Solution

Multiply 5 Â· 7 and multiply i Â· i.

Replace i² with -1.

Multiply.

So, 5i Â· 7i = -35.

5i Â· 7i

= 35 Â· i²

= 35 Â· (-1)

= -35

Note:

We write -35 in the form a + bi like this: 0 + (-35)i

Example 2

Find: 4i(9 - 6i)

Solution

Distribute 4i.

Multiply the factors in each term.

Replace i² with -1.

Simplify.

Write the result in the form a + bi.

So, 4i(9 - 6i) = 24 + 36i.

4i(9 - 6i)

= 4i Â· 9 - 4i Â· 6i

= 36i - 24i²

= 36i - 24(-1)

= 36i + 24

= 24 + 36i

Example 3

Find: (7 - 4i)(10 + 5i)

Solution

Multiply using the FOIL method.

Multiply the factors in each term.

Replace i² with -1.

Combine like terms.

So, (7 - 4i)(10 + 5i) = 90 - 5i.

(7 - 4i)(10 + 5i)

= 7 Â· 10 + 7 Â· 5i - 4i Â· 10 - 4i Â· 5i

= 70 + 35i - 40i - 20i²

= 70 + 35i - 40i - 20(-1)

= 70 + 35i - 40i + 20

= 90 - 5i