College Algebra with Current Interesting Applications and Facts
by Acosta & Karwowski, Kendall Hunt, 2012
Textbook Supplement
Contents
A.
Complex Numbers
2
B.
Distance Formula, Midpoint Formula, and Circles
9
C.
Synthetic Division
23
Topic Expansion from Content Already in Textbook
D.
More on Rational Functions
33
E.
Non-Linear Systems of Equations
47
F.
Other Types of Equations
61
G.
Solving Polynomial and Rational Inequalities
71

2
Section A:
Complex Numbers
Suppose we wish to solve the equation
x
2
=
−
25. Since we know that the square of a real number
is either zero or positive, there is no real number that would satisfy this equation. To solve this
problem, mathematicians created a number system that is based upon a new number: the
imaginary unit,
commonly referred to as "
i
."
Little Facts
: It is said that the name "imaginary number" was originally coined by René Descartes in the
seventeenth century as a derogatory term, because at that time such numbers were regarded by some as fictitious or
useless. Swiss mathematician Leonhard Euler introduced the letter "i" to represent the square roots of negatives in
1777. In modern times these numbers have essential, concrete applications in math, physics, electrical engineering,
and many other scientific and related areas. Sources:
The Imaginary Unit
The
imaginary unit
,
i
, is defined with the following properties
i
=
√−1
and
i
2
=
√−1
•
√−1
=
−
1
We can use the imaginary unit to rewrite the square root of a negative number.
Rewriting the Expression
√−
If
a
> 0
,
then
√−
=
i
√
Let us rewrite
√−25
as a product of a real number and
i
:
√−25
=
(−1)(25)
=
i
√25
= 5
i
We can check the answer by squaring 5
i
:
(5
i
)
2
= 5
2
i
2
= (25)(
−
1)
=
−
25
Example 1
Rewrite
√−28
in terms of
i
, and simplify if possible.
Solution
√−28
=
i
√28
=
i
√4
•
7
= 2
i
√7
Note:
It is also acceptable to write an expression like 2
i
√7
as 2
√7
i,
but we must be sure to write the "
i
"
outside the radical symbol. To avoid being read as being under the radical, we generally write the answer
with "
i
" in front of the radical.
When mathematicians added a real number to multiples of imaginary units, the set of
complex
numbers
was formed.
Complex Number
A complex number is one of the form
a
+
bi
, where
a
and
b
are real numbers.
In a complex number, we call
a
the real part and
b
is
the imaginary part.
Two complex numbers
a
+
bi
and
c
+
di
are equal if and only if
a
=
c
and
b
=
d
.
Any real number,
a
, can be written as a complex number as
a
+ 0
i
. In this case,
b
= 0. If a
complex number has
b
≠
0, then we call
a
+
bi
, an
imaginary number (nonreal complex