Chapter 3: Polynomial Functions
3.1 Characteristics of Polynomial Functions
In its most basic definition, a polynomial function is one where 𝑓(𝑥) is equal to a
polynomial. There are, however, some restrictions:


Linear and quadratic functions, which we have seen before, are polynomial functions.
Others might include:
Some examples of non-polynomial functions are:
Some definitions, before we move on;
Degree End behaviour Leading coefficient The official mathematical definition of a polynomial function is that it is of the form 𝑓(𝑥) =
𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + 𝑎𝑛−2 𝑥 𝑛−2 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 , where 𝑛 is a whole number (and 𝑥 is
a variable).
Note that we arrange the terms by descending degree.
Let us also note that there are no restrictions on the domain of a polynomial function. That
is, for all polynomial functions, the domain is {𝑥|𝑥 ∈ ℝ}. (In practical application, this
might change based on contextual limitations.)
Suppose we have a polynomial function 𝑓(𝑥) of even degree and a positive leading
coefficient. What happens to 𝑓(𝑥) as:
a.
𝑥 approaches ∞?
b.
𝑥 approaches −∞?
Suppose we have a polynomial function 𝑓(𝑥) of odd degree and a positive leading
coefficient. What happens to 𝑓(𝑥) as:
a. 𝑥 approaches ∞?
b. 𝑥 approaches −∞?
Considering that the leading coefficients could also be negative, we can say, without loss of
generality, that:
 If a polynomial function is of even degree,

If a polynomial function is of odd degree,
Finally, let us consider how many times a polynomial function changes direction and how
many 𝑥-intercepts it has.
Linear functions (degree 1) never changed direction and had one 𝑥-intercept. Quadratic
functions (degree 2) changed direction once and had up to two 𝑥-intercepts. In general, we
can say that a polynomial function of degree 𝑛 could change direction up to 𝑛 − 1 times (or
have up to 𝑛 − 1 “bumps”) and have up to 𝑛 𝑥-intercepts.
Homework: textbook pages 114-116, #1-4, 6, 9 (a-c), 12 (a, b)
Chapter 3: Polynomial Functions
3.2 The Remainder Theorem
You know what’s fun? That’s right: polynomial long division. Haven’t you always wanted
to divide polynomials by other polynomials? Well, today’s your lucky day.
Before dividing polynomials, we need to ensure two things about the dividend (the thing
that is being divided):
 There exists a term of every degree from 0 to the polynomial’s degree
 The polynomial is arranged by descending degree
Then we party as we will see in the following examples.
Quick note: when the polynomial 𝑃(𝑥) is divided by the binomial 𝑥 − 𝑎 (𝑎 ∈ ℤ), the result is
𝑃(𝑥)
𝑅
= 𝑄(𝑥) +
𝑥−𝑎
𝑥−𝑎
where 𝑄(𝑥) is the quotient and 𝑅 is the remainder. Note that 𝑥 ≠ 𝑎.
Example: Divide the polynomial 𝑃(𝑥) = 6𝑥 4 + 31𝑥 3 − 27𝑥 − 10 by 𝑥 − 5.
Example: Divide the polynomial 𝑃(𝑥) = 2 − 3𝑥 2 + 2𝑥 3 by 𝑥 + 1.
The only thing that could make polynomial division even more fun would be a way to make
it more efficient. Fortunately, there exists synthetic division, which employs only the
coefficients of the dividend when dividing by a binomial of the form 𝑥 − 𝑎. It requires
fewer calculations, increasing our speed.
We will still have the same requirements as in polynomial long division.
We will tackle this by using the same examples as above.
Example: Divide the polynomial 𝑃(𝑥) = 6𝑥 4 + 31𝑥 3 − 27𝑥 − 10 by 𝑥 − 5.
Example: Divide the polynomial 𝑃(𝑥) = 2 − 3𝑥 2 + 2𝑥 3 by 𝑥 + 1.
When tasked with dividing polynomials, you may use whichever method you prefer, noting
the restriction on the divisor for synthetic division.
The Remainder Theorem
The remainder theorem states that when the polynomial 𝑃(𝑥) is divided by 𝑥 − 𝑎, the
remainder is 𝑃(𝑎).
Example: What is the remainder when 2𝑥 3 − 𝑥 + 4 is divided by 𝑥 + 1?
Example: Is 𝑥 − 3 a perfect divisor of 3𝑥 4 − 24𝑥 2 + 48 − 5𝑥 3 + 20𝑥?
Homework: textbook pages 124-125, #1, 2, 3-5 (a, c, e only), 6-12, 15
Chapter 3: Polynomial Functions
3.3 The Factor Theorem
The remainder theorem states that when polynomial 𝑃(𝑥) is divided by 𝑥 − 𝑎, the
remainder is 𝑃(𝑎). We can extend this to develop the factor theorem.
We know that, for two numbers 𝑚 and 𝑛, when 𝑚 is a factor of 𝑛, the remainder when 𝑛 is
divided by 𝑚 is 0. In the same way, the factor theorem states that 𝑥 − 𝑎 is a factor of
polynomial 𝑃(𝑥) if and only if 𝑃(𝑎) = 0.
Example: Determine if 𝑥 + 1 and 𝑥 − 1 are factors of 3𝑥 3 + 7𝑥 2 − 4.
Given that we can use the factor theorem to discover factors of a polynomial, it would be
handy to have a means to determine what the possible factors are.
The integral zero theorem states that if 𝑥 − 𝑎 is a factor of polynomial 𝑃(𝑥) (which has only
integer coefficients), then 𝑎 is a factor of the constant term of 𝑃(𝑥). This is called the
integral zero theorem as it would provide the zeros of the function 𝑦 = 𝑃(𝑥).
Interpretation: look at the constant term of 𝑃(𝑥). Take all of the integers that are factors of
that constant as 𝑎, and test 𝑃(𝑎) to determine if 𝑥 − 𝑎 is a factor.
Example: Fully factor 2𝑥 3 + 3𝑥 2 − 3𝑥 − 2.
Example: Fully factor 𝑥 3 − 5𝑥 2 − 23𝑥 − 8.
Example: Fully factor 𝑥 4 − 5𝑥 3 − 3𝑥 2 + 45𝑥 − 54.
Homework: textbook pages 133-135, #1, 2-6 (a, c, e only), 7 (a, c only), 11, 13, 14, 16
Chapter 3: Polynomial Functions
3.4 Equations and Graphs of Polynomial Functions
Now that we have the tools to determine the zeros of a polynomial function, we can
determine where the function is positive and negative. We can be assisted through use of a
sign graph, where we plot critical values. For a polynomial function, the critical values are
the zeros.
With this information, we can create a decent sketch of the function.
Example: Create a sign graph to determine where the function 𝑓(𝑥) = 𝑥 4 + 2𝑥 3 − 𝑥 2 − 2𝑥
is positive and negative. Sketch the graph of 𝑦 = 𝑓(𝑥).
When we worked with quadratics, we noted that although a quadratic function could have
two zeros, it could also have one (or none, but we’ll pay little mind to that here). When a
quadratic function had only one zero, it had a factor that was taken to the power of two,
such as 𝑦 = 𝑥 2 or 𝑦 = (𝑥 − 2)2 . This leads us to the concept of multiplicity.
Suppose polynomial function 𝑓(𝑥) has a factor 𝑥 − 𝑎 that is repeated 𝑛 times. We then say
that the zero 𝑥 = 𝑎 is a zero of multiplicity 𝑛.
Example: List the zeros of the function 𝑓(𝑥) = 𝑥 2 (𝑥 + 1)(𝑥 − 1)3 (𝑥 + 2) and state the
multiplicity of each.
A function’s behaviour around each of its zeros depends on the multiplicity of that zero.
Example: Sketch the graph of 𝑓(𝑥) = 𝑥 3 (𝑥 − 4)2 (𝑥 + 2).
When asked to determine the equation for a polynomial function, we should note the zeros
and their multiplicity, as well as the 𝑦-intercept of the function.
Example: Determine a possible equation for the polynomial function shown below.
If we are asked to graph a function (rather than just sketch it), we require a higher degree
of accuracy. In those cases, we can create a sketch as rough work, but also use the equation
of the function to determine specific coordinates so that we can determine precisely where
the crests and troughs are.
Homework: textbook pages 147-151, #1-5, 7, 9 (a, c, e), 10 (a, c), 14, 19, 22 (a, c)
Chapter 3: Polynomial Functions
Review & Questions from June 2014 Provincial
Textbook review:
Pages 153-154, #1, 2, 4, 5, 7, 8, 10, 11, 13-15
Questions from June 2014 Provincial: (5 questions out of 49, 10 marks out of 90)
Question 8 (2 marks)
a. Determine the remainder when 𝑥 4 − 3𝑥 2 + 1 is divided by 𝑥 + 2.
b. Is 𝑥 + 2 a factor of 𝑥 4 − 3𝑥 2 + 1? Explain your reasoning.
Question 16 (2 marks)
Divide (𝑥 3 − 5𝑥 − 4) by (𝑥 + 1).
Question 28 (1 mark)
A sheet of paper 12 cm long and 8 cm wide is used to make a
box with no lid. Equal squares of side length 𝑥 cm are cut from
each of the corners and the sides are folded up to make the box.
Which of the following expresses the volume of the box?
a. 𝑉(𝑥) = 𝑥(12 + 𝑥)(8 + 𝑥)
b. 𝑉(𝑥) = 𝑥(12 − 𝑥)(8 − 𝑥)
c. 𝑉(𝑥) = 𝑥(12 + 2𝑥)(8 + 2𝑥)
d. 𝑉(𝑥) = 𝑥(12 − 2𝑥)(8 − 2𝑥)
Question 42 (2 marks)
Write the equation for 𝑓(𝑥) that satisfies all of the following conditions:
 𝑓(𝑥) is a polynomial function of degree 4
 𝑓(𝑥) has a zero at 2 with a multiplicity of 3
 𝑓(𝑥) has a zero at −5
 𝑓(𝑥) has a 𝑦-intercept of 80
Question 45 (3 marks)
Sketch the graph of 𝑓(𝑥) = (𝑥 − 1)2 (𝑥 + 2)3 . Label the 𝑥-intercepts and the 𝑦-intercepts.