Precalculus I: College Algebra
Section 4.1: Polynomial Functions
Math 141
Math 141
Polynomial Functions
A polynomial of degree n is a polynomial function of the form
f (x) = an xn + an−1 xn−1 + an−2 xn−2 + · · · + a2 x2 + a1 x + a0
where n is a non-negative integer exponent and an 6= 0.
• n can only take on values of n = 0, 1, 2, 3, 4, . . .
• a0 , a1 , a2 , . . . , an−1 , an are the coefficients of the polynomial.
• an is leading coefficient (the coefficient associated with the highest degree term).
• a0 is called the constant coefficient.
Example: For the following, if the function is a polynomial, state its degree, leading coefficient and constant
coefficient. If it is not a polynomial, please state why?
1. f (x) = −5x3 + 2x − 10
This is a polynomial since each
 exponent is a non-negative integer.
 degree: 3
leading coefficient: − 5
The highest degree term is −5x3 while constant is -10, so

contant coefficient: − 10
3
2. g(x) =
This is NOT a polynomial because we can re-write the function as g(x) = 3x−1 , a
x
polynomial can not have a negative exponent.
√
h(x) = x2 − 2x + x
This is NOT a polynomial due the radical term which can be re-written as
3. √
x = x1/2 , a polynomial can not have any exponent with fractions.
√
4. j(x) = −3x + 2 x2 − 25 x4
This is a polynomial since
 each exponent is a non-negative integer.
 degree: 4
leading coefficient: − 52
The highest degree term is − 25 x4 while constant is 0, so

contant coefficient: 0
5. k(x) = 3x2 (x + 1)2 (3 − 2x)3
This is a polynomial since it is the product of three polynomial factors.
To determine the Highest degree term we only have to keep the largest variable term within each factor,
then multiply the remaining expressions.

 degree: 7
leading coefficient: − 24
Highest degree term: 3x2 (x)2 (−2x)3 = −24x7 −→

contant coefficient: 0
Graphing a polynomial function is a necessity in this course and all future courses. In calculus, you will
learn that the graph of a polynomial function is always a smooth curve (no sharp corners or cusps) and
continuous (no holes or gaps).
Investigate the graphs of the following functions known as Power Functions: f (x) = Axn where n is any
non-negative integer.
• Consider the first few EVEN powers of n −→ f (x) = x2 , x4 , x6 (these are EVEN functions)
Behavior of the function about the x-intercept/Real zero
As the power increase the graph FLATTENS near the real zero.
Functions behavior as |x| → ∞
Basically, what is y doing as x grows without bound?
As x → ∞ notice y tends to infinity, so , y → ∞ as x → ∞.
As x → −∞ notice y tends to infinity, so , y → ∞ as x → ∞.
Even degree with Positive Leading Coefficient Arms Go Upwards.
What if the leading coefficient is NEGATIVE ?
If the leading coefficient is negative the graph will FLIP over
the x-axis and the arms of the graph will be going DOWNWARDS.
As x → ∞ notice y tends to - infinity, so , y → −∞ as x → ∞.
As x → −∞ notice y tends to - infinity, so , y → −∞ as x → ∞.
• Consider the first few ODD powers of n −→ f (x) = x3 , x5 , x7 (these are ODD functions)
Behavior of the function about the x-intercept/Real zero
As the power increase the graph FLATTENS near the real zero.
Functions behavior as |x| → ∞
Basically, what is y doing as x grows without bound?
As x → ∞ notice y tends to infinity, so , y → ∞ as x → ∞.
x → −∞ notice y tends to - infinity, so , y → −∞ as x → −∞.
Odd degree with Positive Leading Coefficient Arms Opposite
DOWN to LEFT and UP to RIGHT.
What if the leading coefficient is NEGATIVE ?
If the leading coefficient is negative the graph will FLIP over
the x-axis and the arms of the graph will FLIP
As x → ∞ notice y tends to - infinity, so , y → −∞ as x → ∞.
x → −∞ notice y tends to infinity, so , y → ∞ as x → −∞.
Odd degree with Negative Leading Coefficient Arms Opposite
UP to LEFT and DOWN to RIGHT.
Example: f (x) = 2 − x3 . Behaves as Odd degree with Negative Leading Coefficient (−x3 ), its arms
will go UP to LEFT and DOWN to RIGHT. See graph below.
Example: f (x) = (x2 − 4)(x − 1)(x + 5). Behaves as an EVEN degree with Positive Leading Ceofficient
(x ), its arms will both GO UPDWARDS. See graph below.
4
Summarize the End Behavior of a Polynomial Function.
Look at the degree of the polynomial and the leading coefficient to determine the behavior of the polynomial
as x gets large. The polynomial f (x) = an xn + an−1 xn−1 + an−2 xn−2 + · · · + a2 x2 + a1 x + a0 has the same
end behavior as the monomial y = an xn . The principle that we have just stated is easy to apply because
we already know how to graph functions of the form y = A xn . The following statements summarize this
behavior. (In general, the highest degree term dominates over all other terms.)
• if the degree of the polynomial is even and the leading coefficient is positive, then both ”arms”
will point in the positive direction as |x| gets large: y → ∞ as |x| → ∞.
• if the degree of the polynomial is even and the leading coefficient is negative, then both ”arms”
will point in the negative direction as |x| gets large: y → −∞ as |x| → ∞.
• if the degree of the polynomial is odd and the leading coefficient is positive, then both ”arms”
will point in opposite directions and as x gets large, the arm points up
y → ∞ as x → ∞ and y → −∞ as x → −∞.
• if the degree of the polynomial is odd and the leading coefficient is negative, then both ”arms”
will point in opposite directions and as x gets large, the arm points down
y → −∞ as x → ∞ and y → ∞ as x → −∞.
The Behavior of a Polynomial Function Near an x-intercept/Real zero.
*** The x-intercepts/Real zeroes play a major role in the graph of a polynomial function. At the xintercepts/Real zeroes the graph must either cross the x-axis or touch the x-axis. Also, in between
consecutive real zeroes the graph will either be above the x-axis or below the x-axis.
Example: Take the following function in factored form: f (x) = (x + 2)(x − 3)2
Find the values of the real zeroes (x-intercepts). These are the values where f (x) = 0.
x + 2 = 0 → x = −2
(x − 3)2 = 0 → x = 3
Find the y-intercept (Set x=0), so y = f (0) = (2)(−3)2 = 18.
notice the graph Crosses thru x = −2
but the graph Touches at x = 3
The function above is a perfect example of a 3rd degree polynomial with a positive leading coefficient.
The real zero of x = −2 is said to have multiplicity of 1 while the real zero of x = 3 is said to have a
multiplicity of 2. Multiplicity refers to the number of times the zero occurs.
Note that the factors of the polynomial function were instrumental in determining the real zeroes. Likewise,
if you know the real zeros (and perhaps the multiplicity) you will know the factors of the polynomial and
can construct some polynomial function modeling the given information.
Example: Find a 4th degree polynomial whose zeroes are 1 with a multiplicity of 2, -2 and -4.
Zeroes
Factors
x=1
multiplicity 2 (x − 1)2
x = −2 multiplicity 1 x + 2
x = −4 multiplicity 1 x + 4
Possible Polynomial Function: f (x) = A(x − 1)2 (x + 2)(x + 4), A is a nonzero constant.
Let f (x) be polynomial and suppose that (x − a)n is a factor of f (x). [Furthermore, assume that none
of the other factors of f (x) contains (x − a).] Then, in the immediate vicinity of the x-intercept at a, (here
a is called a zero of multiplicity n) then the graph of y = f (x) closely resembles that of y = A(x − a)n .
The principle that we have just stated is easy to apply because we already know how to graph functions of
the form y = A(x − a)n . The next example shows how this works.
g(x) = x(x + 1)(x − 3)2
Near real zero at x = 3, the graph has the same general
shape as that of y = A(x − 3)2
In other words, graph looks quadratic at x = 3,
graph looks linear at x = −1 and x = 0.
h(x) = x(x + 1)(x − 3)3
Near real zero at x = 3, the graph has the same general
shape as that of y = A(x − 3)3
In other words, graph looks cubic at x = 3,
graph looks linear at x = −1 and x = 0.
An Approach to Graphing Polynomial Functions
1. Factor the polynomial, if possible.
2. Find the y-intercept, set x = 0.
3. Find the real zeroes, set y = 0 and solve for x.
4. Determine the behavior near the real zero.
• Even Multiplicity → graph touches at the real zero.
– Sign of f (x) does not change sign on either side of the real zero.
• Odd Mulitplicity → graph crosses at the real zero.
– Sign of f (x) changes sign from one of the real zero to the other
5. Determine the end behavior.
Look at the degree of the polynomial and the leading coefficient to determine the behavior as x gets
large.
6. Plot points as necessary to determine the general shape of the graph. But all we really need is one
point in any interval to determine whether the graph is above or below the x-axis in that interval, then
we can fill in the rest based on the knowledge of the multiplicity of the real zero.
Example:
Sketch the function, f (x) = (x − 4)(x + 2)2
y-intercept: y = f (0) = (−4)(2)2 = −16
Real zeroes:
x−4=0
x=4
crosses (linear)
(x + 2)2 = 0 x = −2 touches (quadratic)
End behavior x3 Down Left and Up Right
Example:
Sketch the function, f (x) = −5x2 (x + 2)3 (x − 2)
y-intercept: y = f (0) = −5(0)2 (1)3 (−2) = 0
Real zeroes:
−5x2 = 0
x=0
touches (quadratic)
(x + 2)3 = 0 x = −2 crosses (cubic)
x−2=0
x=2
crosses (linear)
2
3
End behavior −5x (x) (x) = −5x6 Both arms down
Example: Consider the following graph of a polynomial function f (x). Assume the polynomial has no
turning points beyond those shown.
1. Is the leading coefficient positive or negative? why?
The leading coefficient must be NEGATIVE due to both arms going down.
2. Can the degree of this polynomial ever be odd? why?
No, it can never be odd due to the arms of the graph going down.
3. Find the
zeroes
x = −5
x = −2
x=3
x=6
real zeroes of the polynomial and notice the multiplicity.
multiplicity
Factor
crosses → mult. of 1 x + 5
touches → mult. of 2 (x + 2)2
crosses → mult. of 1 x − 3
touches → mult. of 2 (x − 6)2
4. Find a possible polynomial that could model this graph in factored form.
f (x) = −(x + 5)(x + 2)2 (x − 3)(x − 6)2