Module 11 Lesson 3
Polynomial Functions - Part I
Definition of Polynomial Functions:
Let ‘n’ be a nonnegative integer.
Let all of an , an1,..., a2 , a1, a be Real Numbers with an  0
The function: f ( x )  an x n  an 1 x n 1  ...  a2 x 2  a1 x  a0
is called a Polynomial Function of x with Degree n
The degree is the value of the largest exponent of the function. (You should first write
the polynomial in descending order by exponent!) The leading coefficient is the
coefficient of the term containing the degree.
Here are some polynomial functions:
f ( x )  3x 3  12
f ( x)  9
Here are some functions which are NOT polynomials:
f ( x)  x 2
f ( x)  3x101
f ( x) 
f ( x) 
1
x
x3
Why are these NOT polynomials? Remember that polynomial functions have x-terms
with non-negative integers as their exponents.
In the list of functions which were NOT polynomial, they can be rewritten like this:
f ( x )  x 1
1
f ( x)  ( x  3) 2
The first doesn’t have a non-negative exponent.
The second doesn’t have an integer exponent.
The degree can sometimes be found from the graph of a polynomial function as well.
When graphed, a polynomial has, at most, a degree of one more than there are turns.
For example:
y = x is a polynomial of degree 1.
The graph has no turns.
y = x2 is a polynomial of degree 2.
The graph has one turn.
y = x3 is a polynomial of degree 4.
The graph has two turns.
y = x4 - 2x2 is a polynomial of degree 4.
The graph has three turns.
However, as in most things, there are exceptions to the rule.
Case in point: y = x8 – x4 – when graphed, this polynomial looks to only have 4 turns.
Sometimes the turns are not visible, no matter how much we zoom in. So be careful
when trying to name a polynomial from the graph alone.
The Special Polynomial:
One last special case: y = 0
This is a polynomial! It is called the zero polynomial. It would have degree zero as the x
term is not written. Remember that x0 = 1.
Naming Polynomial Functions:
We generally name polynomial functions in two different ways – by degree and by
number of terms:
0
1
2
3
4
Example:
Degree
Constant
Linear
Quadratic
Cubic
Quartic
0
1
2
3
4
Number of Terms
n/a
Monomial
Binomial
Trinomial
Polynomial
Name these polynomials by degree and number of terms:
a)
3x 2  x  8
The degree is 2, there are 3 terms, so this is a
quadratic trinomial.
b)
x 5
The degree here is 1, there are 2 terms, so this is a
linear binomial.
c)
5x 4
The degree is 4, there is one term, so this is a
quartic monomial.
Any polynomial with 4 or more terms is referred to as a polynomial. There are more
names for higher degree polynomials, but you won’t need those for this course.
End Behavior of Polynomial Functions
Polynomial functions can be grouped into 4 different categories of end behavior. End
behavior is the name for “where does the function head at its extremes?”
We can look at four simple functions to see those types.
The functions are:
x 2 ,  x 2 , x 3 , and  x 3
Let’s take a look: (don’t worry about those letters yet)
x2
EP
 x2
x3
OP
 x3
EN
ON
The difference in these four functions is where they head off to at the ends.
Read through this section slowly and make sure you understand it!!!
The first function, x 2 , goes up at both ends. This means that as the x-values become
very large or very small, the function increases toward infinity.
As the values for x get large (they head off to the right of the graph), we say they go to
infinity. As values of x get small (they head off to the left of the graph), we say they go
to negative infinity. When the y-values head up, they’re going to infinity, when the yvalues head down, they’re going toward negative infinity.
Here’s the official way to write the end behavior for x 2 :
As
x
x  
and as
y
y
Next, how is  x 2 different? This time as x gets both large and small, the y-values head
down…toward negative infinity.
Here’s the official way to write the end behavior for  x 2 : As
x
x  
and as
y  
y  
For the cubed functions, you can see that the y-values are heading in opposite directions
of each other. For the squared functions, they were either both up or both down.
For the function x 3 , as x gets large, the y-values get large. As x gets small, the y-values
get small.
Here’s the official way to write the end behavior for x 3 :
As
x
x  
and as
y
y  
Finally,  x 3 has the opposite happening. As x gets large, the function is heading down
toward negative infinity. As x gets small (off in the left direction), the function is
heading up toward infinity.
x
x  
Here’s the official way to write the end behavior for  x 3 : As
and as
y
y  
Every polynomial function falls into one of these four categories. All you need to look at
to determine which one is the first term.
The first term should tell you the degree of the polynomial. If the function has a degree
that is even, the behavior will follow one of the x-squared functions. If the degree is an
odd-numbered value, the behavior will follow one of the x-cubed functions. Now you’ve
got it narrowed down to two choices.
Next, is the leading coefficient positive or negative? That will tell you which of those
two it is.
Even Degree & Positive Leading Coefficient follows the behavior of:
Even Degree & Negative Leading Coefficient follows the behavior of:
Odd Degree & Positive Leading Coefficient follows the behavior of:
Odd Degree & Negative Leading Coefficient follows the behavior of:
x2
 x2
x3
 x3
Look at the graphs above. See the E/O/P/N designations? That’s what they mean.
I can’t stress enough that the only part of the function you need to look at is the term
carrying the degree. Even a function like this:
f ( x )  x5  1,000,000,000,000 x 2
will still follow the Odd/Positive behavior of x 3
Why? Because as x gets truly, truly large and close to infinity, the first term of the
polynomial will far, far, far outweigh anything else in the function. In the function
above, sure values of x that are 1, 10, 100 are greatly affected by that crazy term…but
what about when x is equal to 1,000,000,000,000,000,000,000,000,000,000,000? What
happens when you take that to the 5th power? It’s a LOT bigger than the contribution
made by that second term.
Fill in the chart below with the examples above and include the end behavior of each type
of function. This table will help you organize the information given above, more easily.
Degree 
Leading
Coefficient 
Even
Odd
Positive
Negative
Example:
Using your chart above, describe the end behavior of these polynomial functions.
a)
f ( x)  7 x 2  2 x  5
b)
g ( x )  12 x 3
c)
h( x )  400 x 3  100,000 x
Answers:
Example:
a)
f(x):
As
x
x  
and as
y  
y  
b)
g(x):
As
x
x  
and as
y
y  
c)
h(x):
As
x
x  
and as
y
y  