Section 3.1 Polynomial Functions and Models
A polynomial function is a function of the form f
 a n x n  a n

1 x n

1    a
1 x
 a
0
, where n is a whole number {i.e. 0, 1, 2, 3, 4, …} and a n
, a n

1
,  , a
1
, a
0
are real number coefficients. The number n is called the degree (or order ) of the polynomial (i.e. the degree of a polynomial function is the largest power of x that appears). The domain is the set of all real numbers.
The standard form of a polynomial is to list its terms in descending order with respect to the value of the exponents on the variables.
The graph of the polynomial function f
 a n x n  a n

1 x n

1    a
1 x
 a
0
has several characteristics. First, its end behavior (behavior for large values of x , either positive or negative) resembles the graph of the power function f
 
 a n x n
. Second, the graph is smooth , which means the graph contains no sharp corners or cusps. Third, the graph is continuous , which means the graph has no gaps or holes and can be drawn without lifting pencil from paper.
Zeros of a Polynomial Function and Their Multiplicity
If f is a polynomial function and r is a real number for which f

0 , that is, if r is a
(real) zero of f then a.
r is an x -intercept of the graph of f , and b.
 x
 r

is a factor of f .
If
 x
 r
 m
is a factor of a polynomial and
 x
 r
 m

1
is not a factor of f , then r is called a zero of multiplicity m of f.
If r is a zero of even multiplicity then the graph of f touches the x -axis at r and the sign of f does not change from one side to the other side of r .
If r is a zero of odd multiplicity then the graph of f crosses the x -axis at r and the sign of f changes from one side to the other side of r .
Turning Points
The points at which a graph changes direction are called turning points . These points are called local maxima or local minima in calculus and their location is important for many real world applications. How to find them algebraically will be discussed in calculus. For our class it is sufficient just to know how many, at most, that there are. The following theorem tells us the maximum possible turning points a polynomial function can have.
Theorem
If f is a polynomial function of degree n , then f has at most n

1 turning points.
If the graph of a polynomial function f has n

1 turning points, the degree of the f is at least n .
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Section 3.2 Properties of Rational Functions
A rational function is a function f that is a quotient of two polynomials. In other words, f
 
 p
  q
  is a rational function if p
 
and q
 
are polynomials and if q
 
is not the zero polynomial.
The domain of f
 
consists of all real numbers x for which q
 

0 , since division by zero is undefined.
Consider the rational function f

1 x
. Its graph is below.
Notice how the graph looks like it has been ripped apart near x

0 . Not all graphs of rational functions separate like this. When they do we call the line of separation an asymptote .
There are three major types of asymptotes.

Vertical asymptotes which are also called poles

Horizontal asymptotes , and

Oblique asymptotes
2

We need to determine what asymptotes there are for a given rational function in order to graph it.
Vertical asymptotes of a rational function occur at any values of x that make the denominator equal to zero, provided that all common factors, other than constants have been eliminated.
Example 1: Find the vertical asymptotes of the following functions. a. f

1
There are no factors, other than 1, common to both the numerator and the x denominator. So, set the denominator equal to zero and solve for x . So the vertical asymptote is the vertical line x

0 . b. f ( x )
 x 2 x
 x

2

6
. To find out if there are common factors, I need to completely factor the numerator and denominator. This produces the following. f
   x
 x
2

 x
2

3

 x

3 . So, there is not a vertical asymptote for this rational function. c. f ( x )
 x
2 x

4
. Again, I need to completely factor both the numerator and denominator. This produces the following. f . So, the numerator and denominator have no common factors other than 1. Hence there are two vertical asymptotes and they are the vertical lines x
 
2 .
Horizontal asymptotes of a rational function occur as follows.

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always y

0 .

If the degree of the numerator and the denominator are the same, then the horizontal asymptote is determined by the ratio of the coefficients of the leading terms. For example, if f
 ax n bx n
 
 
, where ax n
and bx n
are the leading terms of the numerator and denominator respectively, then the horizontal asymptote occurs at y
 a
. b

If the degree of the numerator is greater than the degree of the denominator, no horizontal asymptote occurs.
3

Example 2: Find the horizontal asymptotes of the following functions. a. f
 

1
. The degree of the numerator is 0 and the degree of the denominator is 1. x
Hence the horizontal asymptote is y

0 . b. f ( x )
 x
2  x

6 x

2
. The degree of the numerator is 2 and the degree of the denominator is 1. Hence there is no horizontal asymptote. c. f ( x )
 x
3 x
2
2 
4
. The degree of the numerator and denominator is 2. Hence the horizontal asymptote is y

3
1

3 d. f

7 x
3
2 x
3

9 x
2 
1

3 x

10
. The degree of the numerator and denominator is 3. Hence the horizontal asymptote is y

7

3 .
5 .
2
An oblique asymptote occurs only when the degree of the numerator is greater than the degree of the denominator. If the difference of the degrees is only one, the oblique asymptote is a line (not vertical or horizontal). This is the only type of oblique asymptote we will discuss. However, please notice that a rational function cannot have both a horizontal and oblique asymptote.
The oblique asymptote is found by dividing the rational expression through the process of long division. The quotient obtained is the line which makes the oblique asymptote.
Example 3: Determine the oblique asymptotes of the following functions. a. f ( x )

2 x
2 x

3 x

2

1
. Performing the long division we obtain the following. x

2 2 x
2


2 x
2
2 x

3 x

1

1

4 x

 x x


1

2

So the oblique asymptote is y

2 x

1
1
4

b. f
 x
8 x
3
2 
4
Performing the long division we obtain the following. x
2
8 x

0 x


4

8
8 x
3 x
3 

0 x
0 x
2
2 

0 x
32 x


0
So the oblique asymptote is y

8 x .

32 x

0 c. f
 x
3  x
2
2 x
2 
3 x

25
Performing the long division we obtain the following. x
2 
0 x

25

 x
3 x
3 
2 x
2

0 x
2 x

3 x


2
0

25 x


2

2 x x
2
2


22 x
0 x


0
50

22 x

50
So the oblique asymptote is y
 x

2
Review of Polynomial Long Division
Let f ( x ) g ( x )
be a rational function where the degree of f ( x ) is more than the degree of g ( x ) .

Then perform the long division g ( x ) f ( x ) by determining what you must multiply the leading term of g ( x ) by to obtain the leading term of f ( x ) .

Continue the division operation until the remainder is 0, or until the degree of the remainder is less than the degree of the divisor. If you do not obtain a remainder of 0, either you made a mistake in your division or g ( x ) is not a factor of f ( x ) .
Note : To keep like terms in columns when dividing f ( x ) , use coefficients of 0 for missing terms.
One can always check their work of polynomial division by the following. f ( x )
 divisior
 quotient
 remainder
5

Example 4: Find the quotient and remainder for x
3 
7 x
2 
13 x

3 x

2
. x
2 
5 x

3 x

2

 x
3 x
3


7 x
2
2 x
2


13 x

3




5
5 x x
2
2


13 x
10 x


3

3 x x


3

3 6

9
So the quotient is x
2 
5 x

3 and the remainder is R = 9.
6

Section 3.3 The Graph of a Rational Function
To graph a rational function f
 
 q p
 
  , where p
 
and q
 
have no common factors:
1.
Find the vertical asymptotes and sketch them into the graph with a dotted line.
2.
Find the horizontal or the oblique asymptote, if there is one, and sketch it into the graph with a dotted line.
3.
Find and plot the zeros of the function. The zeros are the values of x that make the numerator equal to zero. In other words, the zeros of f ( x )
4.
Find and plot f
 
. This is the y -intercept of the function.
are the zeros of p ( x ) .
5.
Find other function values to determine the general shape of the function. Then draw the graph.
Note: The graph of a rational function never crosses a vertical asymptote. However, it may or may not cross a horizontal or oblique asymptote. In addition, the end behavior of the graph will approach the horizontal or oblique asymptote.
Examples : Graph each of the following functions. a. f ( x )
 x
2
8

4
7

b. f ( x )

3
3
 x c. f ( x )
 x
2
2 x
 x

2
2 
1
8

d. f
 x
3 x
2

2 x
2 
15 x

5 x

14
9

Section 3.4 Polynomial and Rational Inequalities
A polynomial inequality is written by replacing the equal symbol in a polynomial equation with one of the following order symbols:

,

,

, or

. A polynomial inequality in standard form is written in one of the following ways.




P ( x )

0
P ( x )

P ( x )

0
0
P ( x )

0 where P ( x ) is a polynomial function.
To Solve Polynomial Inequalities
1.
Write in standard form.
2.
Set P ( x )

0
3.
Factor P ( x ) and solve for x .
4.
Using these values of x , divide the number line into intervals.
5.
Pick a value inside of each interval and see if it makes the original inequality true.
6.
The solution set is all intervals containing values which make the inequality true.
7.
Use interval notation to describe the solution set.
A rational inequality is written by replacing the equal symbol in a rational equation with one of the following order symbols:

,

,

, or

. A rational inequality in standard form is written in one of the following ways.




R ( x )

R ( x )

0
0
R ( x )

R ( x )

0
0 where R ( x ) is a rational function.
To Solve Rational Inequalities :
1.
Write in standard form.
2.
Find all values of x which make the rational function undefined.
3.
Set R ( x )

0 and solve for x .
4.
Using the values of x found in steps 2 and 3, divide the number line into intervals.
5.
Pick a value inside of each interval and see if it makes the original inequality true.
6.
The solution set is all intervals containing values which make the inequality true.
7.
Use interval notation to describe the solution set.
10

Section 3.5 The Real Zeros of a Polynomial Function
Division Algorithm for Polynomials
If f
 
and g
 
denote polynomial functions and if there are unique polynomial functions q
 
and
  g
 
is not the zero polynomial, r x such that f g
 
   
 r g
 
  or f degree less that that of the divisor ,
 
, where
  r
 
is either the zero polynomial or a polynomial of g x . In the preceding equations, q x is the quotient , and r
 
is the f
 
is the dividend , remainder . g
 
is
Remainder Theorem
Let f be a polynomial function. If f
 
is divided by x
 c , then the remainder is f
 
.
We can use this theorem to determine the remainder of polynomial division without performing the division.
Factor Theorem
Let f be a polynomial function. Then x
 c is a factor of f
 
if and only if f
 

0 .
In other words, x
 c is a factor of f
 
if and only if the remainder is zero.
Theorem
A polynomial function cannot have more real zeros than its degree.
Descartes’ Rule of Signs provides information about the number and location of the real zeros of a polynomial function written in standard form.
Descartes’ Rule of Signs
Let f denote a polynomial function written in standard form.
The number of positive real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f
 
or else equals that number less an even integer.
The number of negative real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f
 
or else equals that number less an even integer.
Finding Rational Roots of a Polynomial
If a polynomial possesses a root that is a rational number, we can find it using the following theorem.
11

Rational Root Theorem
Let f ( x )
 a n x n  a n

1 x n

1  a n

2 x n

2    a
1 x
1 a
0 x
0
, where all the coefficients are p integers. Consider a rational number denoted by q
, where p and q are relatively prime p
(having no common factors besides –1 and 1). If q
is a root of f , then p is a factor of a
0 and q is a factor of a . n
This theorem can be translated to say that if f has a rational root, then the possible rational roots are
 factors of factors of a
0
. a n
It is a common practice to find the real zeros, or x -intercepts, of polynomials and to use these points to aid in graphing them.
Steps for Finding the Real Zeros of a Polynomial Function
1.
Use the degree of the polynomial to determine the maximum number of zeros.
2.
Use Descartes’ Rule of Signs to determine the possible number of positive zeros and negative zeros.
3.
If the polynomial has integer coefficients, use the Rational Zeros Theorem to identify those rational numbers that potentially could be zeros. a.
Use substitution, synthetic division, or long division to test each potential rational zero b.
Each time that a zero (and thus a factor) is found, repeat Step 3 on the depressed equation.
4.
In attempting to find the zeros, remember to use (if possible) the factoring techniques that you already know (special products, factoring by grouping, and so on).
Example 1 : Find all the roots (or zeros) of f ( x )
 x
3 
3 x

2
Possible rational roots are

1 ,
1
2 x
 
1 and x

2 are rational roots. Choosing to work with x
 
1 we have
12

x
2 x

1 x
3 
0 x
2

 x
3  x
2

 x

2

3 x

2



 x x
2
2


3 x x


2




2
2 x x


2
2

0
So f ( x )
 x
3 
3 x

2

 x

1
  x
2  x

2

Factoring the quadratic quotient we have, f ( x )
 x
3 
3 x

2

 x

1
 x

1
 x

2

So the roots are x = –1, –1, and 2
Example 2: Find the roots (or zeros) of f ( x )

4 x 3 
3 x

18 .
Possible rational roots are

1 , 2 , 3 , 6 , 9 , 18
1 , 2 , 4 x

3
2
is a root. So 2 x

3 is a factor.
2 x
2 
3 x

6
2 x

3 4 x
3


4 x
3

0 x
2

6 x
2


3 x

18

6

6 x x
2
2


3 x
9 x


18

12

12 x x


18
18

0
So f ( x )

4 x
3 
3 x

18


2 x

3
 
2 x
2 
3 x

6

The quadratic quotient will not factor. Using the quadratic formula we find
13

that the three roots are x

3
2
,

3
 i
4
39
.
We can solve polynomial equations by finding the zeros of the associated polynomial function
14

Section 3.6 Complex Zeros; Fundamental Theorem of Algebra
A variable in the complex number system is referred to as a complex variable .
A complex polynomial function f of degree n is a function of the form f
 
 a n x n  a n

1 x n

1    a
1 x
 a
0
, where a n
, a n

1
,  , a
1
, a
0
are complex numbers, a n

0 , n is a nonnegative integer, and x is a complex variable. As before, a is called n the leading coefficient of f . A complex number r is called a ( complex ) zero of f if f
 

0 .
The Fundamental Theorem of Algebra
Every complex polynomial function f
 
of degree n

1 has at least one complex zero.
This theorem has some interesting results.
1.
Every polynomial function f of degree n , with n

1 , can be factored into n linear factors (not necessarily unique). That is, f ( x )
 a n
 x
 c
1
 x
 c
2
  x
 c n

.
2.
Every polynomial function f of degree n , with n

1 , has at least one root and at most n roots. In fact, these roots are c
1
,  , c n
.
3.
Every polynomial function f of degree n , with n

1 , has exactly n roots if one takes into account multiplicities. (Multiplicity is the number of times a root occurs
4.
when a function is factored completely.)
If a complex number a
 bi , where b

0 , is a root of a polynomial function f with real coefficients, then its conjugate, a
 bi , is also a root. In other words, complex roots always come in conjugate pairs.
5.
If a
 c b , where a and c are rational but b is not a perfect square, is a root of a polynomial function f with rational coefficients, then its conjugate a
 c b , is also a root. In other words, roots that are irrational numbers always come in conjugate pairs.
Notice that there isn’t any condition as to whether or not the roots are real or complex. It is possible that all the roots are real numbers. It is also possible that all the roots are complex numbers. In most cases, there are some real roots and some complex roots.
However, there are exactly n roots when you take into account multiplicities.
15