Intermediate Algebra (11th edition)

(c)

FOIL

Multiply.

= 17 - 7 i

= 2 - 7 i+ 15

= 2 - 7 i- 151 - 12

= 2 - 10 i+ 3 i- 15 i^2

= 2112 + 21 - 5 i 2 + 3 i 112 + 3 i 1 - 5 i 2

12 + 3 i 211 - 5 i 2

478 CHAPTER 8 Roots, Radicals, and Root Functions

NOW TRY
EXERCISE 6
Multiply.

(a)

(b) 17 - 2 i 214 + 3 i 2

8 i 13 - 5 i 2

NOW TRY

The two complex numbers and are called complex conjugates, or

simply conjugates,of each other. The product of a complex number and its conju-

gate is always a real number,as shown here.

1 abi 21 abi 2 a^2 b^2

= a^2 - b^21 - 12

1 a+ bi 21 a- bi 2 = a^2 - abi+abi- b^2 i^2

a+ bi a-bi

Use parentheses

around to

avoid errors.

• 1

For example,

OBJECTIVE 5 Divide complex numbers.The quotient of two complex num-

bers should be a complex number. To write the quotient as a complex number, we

need to eliminate iin the denominator. We use conjugates and a process similar to

that for rationalizing a denominator to do this.

Dividing Complex Numbers

Find each quotient.

(a)

Multiply both the numerator and denominator by the conjugate of the denomina-

tor. The conjugate of is

Factor the numerator.

= 2 +i Lowest terms

=

2912 + i 2

29

=

58 + 29 i

29

=

40 - 16 i+ 45 i- 18 i^2

52 + 22

5  2 i

= 5  2 i= 1

18 + 9 i 215 - 2 i 2

15 + 2 i 215 - 2 i 2

8 + 9 i

5 + 2 i

5 + 2 i 5 - 2 i.

8 + 9 i

5 + 2 i

EXAMPLE 7

13 + 7 i 213 - 7 i 2 = 32 + 72 = 9 + 49 =58.

The product

eliminates i.

In the denominator,

Combine like terms.

• 18 i^2 =- 181 - 12 =18;

1 a+bi 21 a-bi 2 =a^2 +b^2.

Factor first. Then
divide out the
common factor.
NOW TRY ANSWERS

6. (a) 40 + 24 i (b) 34 + 13 i