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M101 COLLEGE AND ADVANCED ALGEBRA jvb
Presented to
the Instructional Materials Development
Committee Bohol Island State University Main
Campus Tagbilaran City
jvb
jvb
TITLE PAGE i TABLE OF CONTENTS ii
1 RADICALS AND RATIONAL EXPONENTS 1 1.1 Objectives.........
..................... 1 1.2 Radicals........................
....... 1 1.3 Laws of Radicals......................... 2 1.
Operations of Radical Expressions................ 6 1.
Assessment Task 6........................ 10
This chapter discusses the complex numbers and its operations.
Exam ples are given to further understand the concepts.
1.1 Objectives
Add, subtract, multiply and divide complex numbers.
1.2 Complex Numbers
Definition 1.2.1 The set of complex numbers is given by C = {x + yi : x, y
∈ R } such that i
2 = − 1 with operations of addition (+) and multiplication ( · )
defined as follows:
(1) Addition: ( x 1 + y 1 i ) + ( x 2 + y 2 i ) = ( x 1 + x 2 ) + ( y 1 + y 2 ) i; (2)
Multiplication: ( x 1 + y 1 i ) · ( x 2 + y 2 i ) = ( x 1 x 2 − y 1 y 2 ) + ( x 1 y 2 + x 2 y 1 ) i. Example
1.2.2 Add (6 + 4 i ) + (5 − 7 i ).
Solutions:
(6 + 4 i ) + (5 − 7 i ) = (6 + 5) + (4 i − 7 i )
= 11 − 3 i.
Example 1.2.3 Subtract (2 + 5 i ) − (8 + 6 i ).
Solutions:
(2 + 5 i ) − (8 + 6 i ) = (2 − 8) + (5 i − 6 i )
= − 6 − i.
Example 1.2.4 Multiply (8 + 4 i )(6 − 3 i ).
Example 1.2.9 Simplify
6+2 i
1 − 2 i.
Solutions:
6+2 i
1 − 2 i =
6+2 i
1 − 2 i·
1+2 i
1+2 i
=
6+12 i +2 i +4 i 2
1 2 − (2 i ) 2
6+14 i +4( − 1)
1 − 4 i 2
6+14 i− 4
1 − 4 i 2
2+14 i
1 − 4( − 1)
=
2+14 i
1+
=
2+14 i
5
Example 1.2.10 Simplify
− 4
2 −
√ − 9_._
Solutions
:
1+
√ − 4
2 −
√ − 9 =
1+
2 −
√ √
4( − 1) 9( − 1)
√ 4 i 2
2 −
√ 9 i 2
1+2 i
2 − 3 i
=
1+2 i
2 − 3 i·
2+3 i
2+3 i
=
2+3 i +4 i +6 i 2
2 2 − (3 i ) 2
2+7 i +6( − 1) 4 − 9 i 2
2+7 i− 6
4 − 9 i 2
− 4+7 i 4 − 9( − 1)
=
− 4+7 i 4+
=
− 4+7 i 13_._
4
1.3 Assessment Task 7
Direction: Simplify
(2_._ ) (3 + 2 i ) − (5 + 8 i )
(4_._ ) (7 + 4 i )(9 −i )
2+7 i
1+ i
(6_._ ) (2 i− 5)
3
7 8 i
5
4+
√ − 16
BIBLIOGRAPHY
[1] W. L. Hart, College Algebra. D.C. Heath and Company, United State
of America, 1953
[2] R. Larson and R.P. Hostetler, Algebra and Trigonometry 8Ed.
