BGSU 2.3 Polynomial Functions and Their Graphs Math 1300
Definition 1 (A Polynomial Function)
Let n be a nonnegative interger. Let a
0
, a
1
, . . . , a n be real numbers and a n
= 0 . Then we say that f ( x ) = a
0
+ a
1 x + . . .
+ a n − 1 x n − 1
+ a n x n is a polynomial of degree n .
a n is called the leading coefficient of f ( x ) .
Good Properties of polynomial functions:
•
Smooth: rounded curve without sharp corners
•
Continuous: no breaks on the curve
degree n n is even n is odd leading coefficient a n
> 0 a n
< 0 a n
> 0 a n
< 0 left ( x → −∞ ) right ( x → ∞ )
Example 1 Use the leading coefficient test to determine the end behavior of the give function.
a.
f ( x ) = 3 x
4 − 999 x
3
+ 1342 x
2
+ 2 x − 1 b.
g ( x ) = − 0 .
1( x − 3)( x + 2)
2 x
3
Let f ( x ) be a polynomial. We say that a is a zero of f ( x ) if f ( a ) = 0.
Example 2 Find all zeros of the give function.
a.
f ( x ) = − ( x + 5)
2
( x − 1)
3 b.
h ( x ) = x ( x − 5) + 6
PS. More exercises can be found on P.331 on the textbook.
Ying-Ju Tessa Chen
Last modified: September 6, 2014
1

BGSU 2.3 Polynomial Functions and Their Graphs Math 1300
If a polynomial f of degree n can be written by f ( x ) = ( x − a ) k
Q ( x ) where Q ( x ) is a polynomial of degree n − k and a is not a zero of Q ( x ). Then we say that a is a zero with multiplicity k .
Example 3 Find all zeros and their multiplicities of the give function. State whether the graph crosses the x -axis or touches the x -axis and turns around at each zero.
a.
f ( x ) = − x
3
+ 6 x
2 − 9 x b.
g ( x ) = −
1
5 x ( x +
1
2
)
2
( x − 3)
5
Remark 1 If a is a zero of even multiplicity, then the graph touches the x -axis and turns around at a .
If a is a zero of odd multiplicity, then the graph crosses the x -axis at a . If k is the multiplicity of a and k > 1 , then the graph of the function tends to flatten out near a . (Why?)
Theorem 1 (The Intermediate Value Theorem for Polynomials)
Let f :
R
→
R be a polynomial with real cofficients. If f ( a ) f ( b ) < 0 , then there exists a point c between a and b such that f ( c ) = 0 .
Remark 2 In fact, the Intermediate Value Theorem holds for all continuous functions f : I →
R where
I is a closed interval.
Ying-Ju Tessa Chen
Last modified: September 6, 2014
2

BGSU 2.3 Polynomial Functions and Their Graphs Math 1300
Example 4 Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers.
a.
f ( x ) = x
3 − 3 x
2
+ 1 ; between 0 and 1 b.
g ( x ) = − 0 .
5 x
4 − 2 x
3
+ 3 x
2
; between 1 and 2
If f is a polynomial of degree n , then the graph of f has at most n − 1 turing points. (Why?)
•
Study the end behavior of the graph of the function by using the leading coefficient of the function.
•
Find x -intercept by setting f ( x ) = 0 and y -intercept by setting x = 0 in the function f ( x ).
•
Find all zeros and their multiplicities of the function.
•
Check if the function is even or odd or neither.
•
Use the fact ”If f is a polynomial of degree n , then the graph of f has at most n − 1 turning points.” to check the graph.
Remark 3 In fact, the tool we learn here is not enough to sketch a ”correct” curve of a polynomial.
From the above steps, we may miss some turning points. But through the study here, we still get a picture of how the graph looks like.
Ying-Ju Tessa Chen
Last modified: September 6, 2014
3