Chapter Seven: Polynomial Functions
Section One: An Introduction to Polynomials
Thus far we have discussed many types of functions: linear, absolute value, parametric, quadratic,
exponential, and logarithmic. In this chapter we will analyze yet another type of function: the
polynomial function.
A monomial is a number, variable, or the product of a numbers and variables (no adding or subtracting
and no variables in the denominator of a fraction). We referred to monomials in previous sections
simply as terms. A monomial with only a numeral and no variables is called a constant. If a monomial
has a number and variables, the number is called the coefficient of the term.
 x is a monomial with a coefficient of 1
 3y is a monomial with a coefficient of -3


 x4 y
1
1 4
is a monomial with a coefficient of
because we can write it as
x y
3
3
3
 xyz is a monomial with a coefficient of -1
The degree of a monomial can be found by simply adding the powers on the exponents (not the
coefficient).
EX1: Find the degree of the following monomials
a. 2x 4 y 5
b. 5x 2
c. xyz
A polynomial is a monomial or the sum of multiple monomials. We can classify polynomials in two
different ways.
Degree
Name
Number of Terms
Name
0
constant
1
monomial
1
linear
2
binomial
2
quadratic
3
trinomial
3
cubic
4 or more
polynomial
4
quartic
5
quintic
Notice that some of the things that we have studied in previous chapters are polynomials.
EX2: Classify each polynomial by term and degree
a. 5 x  2 x3  2 x 2
b. x5  4 x3  x5  3x 2  4 x3
EX3: Evaluate the polynomial 3x 4  2 x 2  2 x  5 for x  1.5 (Use substitution and a calculator table)
We add and subtract polynomials simply by combining like terms. Remember that a minus sign before
parenthesis changes the sign of everything in parenthesis. Most of the time we write our answers in
standard form or descending order. This means to write the monomial with highest degree first and
then put the rest in descending order afterwards.
EX4: Find the sum or difference
a. 6 x3  3x 2  4  10  3 x  5 x 2  2 x3
b.

5x
2
 
 6 x  11   8 x
3
 x  2

2
EX5: Graph each polynomial function. Tell if the graph resembles more an S , a U, or a W. Then give the
number of turns the graph makes.
a. g ( x)  2 x3  1
b.
h( x)  2 x 4  3x 2  x  2
c.
g ( x)  3x 4  2
Section Two: Polynomial Functions and Their Graphs
In this lesson we will examine the graphs of polynomial functions. A peak or a valley of a graph is known
as a local maximum (plural: maxima) or local minimum (plural: minima). A graph is said to either be
increasing or decreasing between these extreme values.
EX1: Graph and describe the graph: P( x)  2 x3  x 2  5x  6
Polynomial functions are continuous functions. They have no breaks. A discontinuous function would
have breaks or holes.
The end behavior of a function is a description of what happens as x approaches positive infinity and
negative infinity.
If we remember what the graph of y  x 2 and y  x 3 look like, we can use them to find the end
behavior of any polynomial function.
Note: Remember that a negative in front of these functions flip them vertically.
y  x2
y   x2
y  x3
y   x3
By looking at whether the degree is even or odd and by looking at the sign of the leading coefficient, we
can determine the end behavior of any polynomial function.
EX2: Describe the end behavior of each function.
a. g ( x)  2 x5  3x 2  1
b.
c.
h( x)  2 x 4  3x 2  x  2
g ( x)  3x 4  2
We can use the calculator to match a cubic or quartic function to a set of data points the same way we
did for linear and quadratic regressions.
EX3: The table at the right gives the number of students who participated in the ACT
program during selected years from 1970 to 1995. The variable x represents the
number of years since 1960, and y represents the number of participants in
thousands. Find the regression model for the number of students who participated
in the ACT program during the given years. Use the quartic regression model to
estimate the number of students who participated in 1985. Compare the value with
the actual value from the table.
x
y
10 714
15 822
20 836
25 739
30 817
35 945
Section Three: Products and Factors of Polynomial Functions
Factors can be multiplied together using the distributive property and the FOIL method.
EX1: Write the function f ( x)  x  x 1 x  4 x  3 as a polynomial function in standard form
Let’s now reverse the process. We will now review factoring.
1. Look for any common factors that can be pulled out of every term
2. Do we have any special cases
a. Difference of squares
b. Difference or sum of cubes
3. Do we have a trinomial with an “a” that is 1
4. Do we have a trinomial with an “a” that is not 1
5. Last Resort: Try to group and factor
EX2: Factor the polynomial
a. x3  16 x 2  64 x
b. x3  6 x 2  2 x  8
c. x 3  125
d. x3  27
From chapter five, the zero product property told us how to find zeros (or roots or x-intercepts) once we
have factored a function. The Factor Theorem puts this idea into words: If  x  r  is a factor of a
polynomial equation P  x  , then r is a zero of P  x  and r is a solution to P  x   0 ( P  r   0 ).
EX3: Use substitution to determine whether x 1 is a factor of x 3  x 2  5 x  3 (Is 1 a zero?)
Polynomials can be divided by using principles of long division.
EX4: Divide using long division
x3  3x 2  4 x  12
a.
x2
3
b.  x  x 2  4    x 2  x  1
The process of dividing by using synthetic division is much quicker and easier. When we are dividing by a
linear term we can use synthetic division. However, we cannot use it when dividing by terms with
degree greater than one.
EX5: Divide using synthetic division
x3  3x 2  4 x  12
a.
x2
3
b.  x  2 x  6    x  1
EX6: Given that 2 is a zero of P( x)  x3  3x 2  4 , use division to factor the polynomial.
EX7: Given that -3 is a zero of P( x)  x3  13x  12 , use division to factor the polynomial.
The remainder theorem is a nice trick that we can use to evaluate polynomial expressions. The theorem
states that instead of plugging in a value to evaluate the function, we can use synthetic division with the
x-value as the root. The remainder that we have is the same as the evaluated answer.
EX8: Given P( x)  3x3  4 x 2  9 x  5 , find P (2) . (Plug it in and use synthetic division)
Section Four: Solving Polynomial Equations
We can now use factoring and the zero product property to solve polynomial equations
EX1: Solve the equation 5 x3  12 x 2  4 x  0 . Check with a graph.
As we have mentioned before, these three points are roots (or x-intercepts or zeros) of the graph
f ( x)  5x3  12 x 2  4 x
Therefore, by looking at a graph we could figure out the solutions to polynomial equations.
Sometimes a function can have roots with multiplicity. This means that if r is a root, the factor ( x  r )
occurs multiple times. We can sometimes tell by looking at a graph if a multiple root occurs. Graphs that
are tangent to the x-axis will have multiple roots at that x-value.
EX3: Use a graph, synthetic division, and factoring to find all the roots.
a. x3  3x 2  4  0
b. x3  2 x 2  4 x  8  0
EX4: Find the roots by factoring: x 4  5 x 2  6  0 (this resembles a quadratic equation)
Section Five: Zeros of Polynomial Functions
What if we need to find the roots of a polynomial and we don’t have a graph? We need a place to start.
The rational root theorem gives us a list of possible roots of a polynomial function. The theorem says
that if we let p be the constant term of our polynomial and q be the leading coefficient, then all the
combinations of the factors of
p
gives us a list of possible rational roots (this doesn’t give us irrational
q
or imaginary ones).
EX1: Make a list of all the possible rational roots of the polynomial P( x)  8 x3  10 x 2  11x  2
The location principle can also help us find roots without a graph. Let’s say that we plugged 2 into our
function and get a negative number. This would mean that this point is below the x-axis. We then plug 3
into the function and get a positive value. This point is above the x-axis. Since polynomial functions are
continuous functions this means that somewhere between 2 and 3 the graph will cross the x-axis.
We can now use synthetic division along with these theorems to find all rational roots.
EX2: Find all the rational roots of the polynomial P( x)  8 x3  10 x 2  11x  2
We will now use all the tools that we have to find all the real roots
 Factoring
 Synthetic division
 Graphing calculator
 Quadratic formula
 Rational root theorem
 Location principle
EX3: Find all the real roots of the polynomials
a. f ( x)  x3  6 x 2  7 x  2
b.
g ( x)  x3  9 x 2  49 x  145
From example b we see that polynomial functions can also have imaginary roots. The complex conjugate
theorem tells us that imaginary roots come in pairs. If 3  2i is a root, then its conjugate 3  2i will also
be a root.
Also, the Fundamental Theorem of Algebra tells us that we can know how many total roots a polynomial
function will have (total of real and imaginary). The degree of the polynomial is the number of total
roots it will have.
EX4: Write a polynomial function, P, in factored form and in standard form by using the given
information
a. Degree is 3; zeros are -3, 2, and 4; and P(0)  120
b. Degree is 4; zeros are 2, 5, and 3i ; and P (0)  3
(Note: For further studies, Descartes Rule of Signs gives us another tool we can use for finding zeros)