5.1 Polynomial Functions
In this section, we will study the following topics:

Identifying polynomial functions and their degree

Determining end behavior of polynomial graphs

Finding real zeros of polynomial functions
and their multiplicity

Analyzing the graph of a polynomial function
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Definition of Polynomial Function
In polynomial functions, THE EXPONENTS ON THE VARIABLE
CANNOT BE FRACTIONS AND CANNOT BE NEGATIVE.
Definition of Polynomial Function
Let n be a nonnegative integer and let an, an-1, …, a2, a1, a0 be real
numbers.
The following function is called a polynomial function of x with
degree n.
f ( x)  an x  an 1 x
n
n 1
 ...  a2 x  a1 x  a0
2
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Graphs of Polynomial Functions
4 out of 5 math students surveyed preferred polynomial
functions over the next leading brand of function
(because their properties make them easy to analyze).
One of the most important features of polynomial functions
is the fact that their GRAPHS are SMOOTH and
CONTINUOUS. (This fact is very important in Calculus.)
For now, this fact helps us to find the zeros of polynomial
functions and to sketch their graphs fairly well by hand.
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5
Classifying Polynomials
Polynomials are often classified by their degree. The degree of a
polynomial is the highest degree of its terms.
Degree
Name of Polynomial
Function
Example
Zero
f(x) = -3
First
f(x) = 2x + 5
Second
Third
Fourth
Fifth
f(x) = 3x2 –5x + 2
f(x) = x3 – 2x –1
f(x) = x4 –3x3 +7x-6
f(x) = 2x5 + 3x4 – x3 +x2
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Graphs of Polynomial Functions
You have already looked at the graphs of two polynomial
functions in some detail: linear and quadratic.
It is VERY important to have a general idea as to the
shape of the basic polynomial functions.
We can then apply transformations to the graphs to
obtain new functions.
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The Leading Coefficient Test
Just as we were able to tell if a parabola would open up or
down by looking at the sign of “a ” of the quadratic
function, we can use the LEADING COEFFICIENT of
higher degree polynomials to determine if the left and
right ends of the graph rise or fall.
We call this the END BEHAVIOR of the graph of the
function.
As x  , f ( x)  ?
As x  , f ( x)  ?
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Using the Leading Coefficient to Describe
End Behavior: Degree is EVEN
f ( x)  x 4

f ( x)   x 4
If the degree of the polynomial is even and the leading
coefficient is positive, both ends ______________.
As x  , f ( x)  _____
As x  , f ( x)  _____

If the degree of the polynomial is even and the leading
coefficient is negative, both ends ________________.
As x  , f ( x)  _____
As x  , f ( x)  _____
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Using the Leading Coefficient to Describe
End Behavior: Degree is ODD
f ( x)  x 5

f ( x)   x 5
If the degree of the polynomial is odd and the leading
coefficient is positive, the graph falls to the __________ and
rises to the ______________.
As x  , f ( x)  _____
As x  , f ( x)  _____

If the degree of the polynomial is odd and the leading
coefficient is negative, the graph rises to the _________ and
falls to the _______________.
As x  , f ( x)  _____
As x  , f ( x)  _____
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The Leading Coefficient Test
Example
Use the leading coefficient test to describe the end behavior of the
graphs of the following functions:
f ( x)  ( x  3)
2
g ( x)  4 x  x 3  3 x 4  2 x 7
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Zeros of Polynomial Function
A nifty observation about polynomials:
1.
A polynomial of degree n has at most n real zeros
2.
A polynomial of degree n has at most n – 1 turning points.
For instance, a quartic polynomial (4th degree) can have AT MOST
4 real zeros (it may have fewer than 4 real zeros, as we will see in
a later section). Also, it can have AT MOST 3 turning points. Take a
look:
ow!
This quartic function has 4 real zeros and 3
turning points.
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Zeros of Polynomial Function
VERY IMPORTANT INFORMATION ABOUT REAL ZEROS!
Real Zeros of Polynomial Functions
If f is a polynomial function and r is a real number then
the following statements are equivalent.
1.
2.
3.
4.
x = r is a zero or root of the function.
x = r is a solution of the equation f(x) = 0.
(x - r) is a factor of the polynomial f(x)
(r, 0) is an x-intercept of the graph of f.
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Zeros of Polynomial Function
From the graph, we can conclude that:
f
r1
r2
r3
x
r4
1.
r1, r2, r3, r4 are _____________________ of f
2.
x=r1, x=r2, x=r3, x=r4 are ________________
of f(x) = 0.
3.
(x-r1), (x-r2), (x-r3), (x-r4) are _____________
of f(x).
4.
(r1,0), (r2,0), (r3,0), (r4,0) are
_______________________________ of f
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Zeros of Polynomial Function
Example
3
2
f
(
x
)

x

x
 20 x algebraically.
Find all real zeros of
Use your graphing calculator to verify your answers.
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Repeated Zeros
If a polynomial function has a factor in the form (x – r)m where m is a
number greater than one, then we say that x = r is a repeated zero
of multiplicity m.

If m is EVEN, the graph touches the x-axis at x = r, but does not
cross through. (The graph turns back around at that point…the sign
of f(x) does not change.)
f ( x)   x  4 
2
This function has a
double zero at x = 4.
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Repeated Zeros

If m is ODD, the graph crosses through the x-axis at x = r (it
flattens out a little bit around x = r...sign of f(x) changes.)
f ( x)   x  4 
3
This function has a
triple zero at x = 4.
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Is the degree of this polynomial even or odd? ________________
Is the leading coefficient positive or negative? _________________
What is the minimum degree of this polynomial? _______________
For the polynomial, list all zeros and their multiplicities.
f  x   2  x  2  x  1  x  3
3
4
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Using the Zeros to Write a Polynomial
Function
We can work backwards to find a polynomial function with
given zeros.
Example: Write a cubic function with zeros: –3, 0, and 5
For
each zero, write in the form of a factor (x – r)
f(x) =
Multiply
out the factors and write the polynomial in
simplest form (in descending order)
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Many correct answers
For example, there are an infinite number of polynomials of
degree 3 whose zeros are -4, -2, and 3. They can be expressed in
the form:
f  x   a  x  4 x  2 x  3
f  x    x  4 x  2 x  3
f  x   2  x  4 x  2 x  3
f  x     x  4 x  2 x  3
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Using the Zeros to Write a Polynomial
Function
Example
Find a cubic polynomial function with the given zeros: 2,
(Verify on calculator)
2
, 3
3
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Using the Zeros to Write a Polynomial
Function
Example
Find a cubic polynomial function with the given zeros: -5; 1 multiplicity 2
(Verify on calculator)
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Sketching Polynomial Functions by Hand
Follow these steps when sketching polynomial functions:
1.
Determine the BASIC SHAPE of the graph. You can tell by the
degree of the polynomial.
2.
Use the leading coefficient to determine the END BEHAVIOR of the
graph (whether graph falls or rises as x approaches  and - ).
3.
Find the REAL ZEROS of the polynomial, if possible. These will be
the x-intercepts of the graph. Watch out for repeated zeros.
4.
Plot a few ADDITIONAL POINTS by substituting a few different
values of x into the original function. Include some points close to
the x-intercepts.
5.
SKETCH the graph, keeping in mind that graphs of polynomials are
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smooth and continuous.
25
(Continued)
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Sketching Polynomial Functions by Hand
Example
Sketch the graph of
f ( x)  2 x 4  2 x3  12 x 2 by hand.
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Sketching Polynomial Functions by Hand
f ( x)  2 x 4  2 x3  12 x 2
y-axis: count by 4
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Using the graph to write a polynomial function
Example
Write the equation for the cubic function shown in the graph below.
(0, 5)
29
Using the graph to write a polynomial function
Example
Write the equation for the quartic function shown in the graph below.
(0, -3)
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End of Section 5.1
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