Lecture 8 section 3.2 polynomial equations

MATH 107
Section 3.2
Polynomial Functions

2© 2010 Pearson Education, Inc. All rights reserved
Definitions
A polynomial function of degree n is a
function of the form
where n is a nonnegative integer and the
coefficients an, an–1, …, a2, a1, a0 are real
numbers with an ≠ 0.
f x( ) = an xn
+ an-1xn-1
+ ...+ a2 x2
+ a1x + a0 ,

Definitions
A constant function f (x) = a, (a ≠ 0) which
may be written as f (x) = ax0, is a polynomial
of degree 0.
The term anxn is called the leading term.
The number an is called the leading
coefficient, and a0 is the constant term.

Definitions
Degree Name
0 Zero function: f(x)=0
1 linear
2 quadratic
3 cubic
4 quartic
5 quintic

COMMON PROPERTIES OF
POLYNOMIAL FUNCTIONS
1. The domain of a polynomial function is the
set of all real numbers.

2. The graph of a polynomial function is a
continuous curve.

3. The graph of a polynomial function is a
smooth curve.

EXAMPLE 1 Polynomial Functions
State which functions are polynomial functions. For
each polynomial function, find its degree, the leading
term, and the leading coefficient.
f (x) = 5x4 – 2x + 7
Solution

END BEHAVIOR OF POLYNOMIAL FUNCTIONS
Case 1
n Even
a > 0
The graph
rises to the left
and right,
similar to
y = x2.

Case 2
n Even
a < 0
The graph
falls to the left
and right,
similar to
y = –x2.

Case 3
n Odd
a > 0
The graph
rises to the
right and falls
to the left,
similar to
y = x3.

Case 4
n Odd
a < 0
The graph
rises to the left
and falls to the
right, similar
to y = –x3.

EXAMPLE 2
Understanding the End Behavior of a
Polynomial Function
Let
function of degree 3. Show that
P x( ) = 2x3
+ 5x2
- 7x +11 be a polynomial
  3
2P x x
when |x| is very large.
Solution
P x( ) = x3
2 +
5
x
-
7
x2
+
11
x3
æ
èç
ö
ø÷
When |x| is very large
5
x
,
7
x2
and
11
x3
are
close to 0.
P x( ) » x3
2 + 0 - 0 + 0( ) » 2x3
.Therefore,

THE LEADING-TERM TEST
Its leading term is anxn.
The behavior of the graph of f as x → ∞ or as
x → –∞ is similar to one of the following four
graphs and is described as shown in each case.
The middle portion of each graph, indicated by
the dashed lines, is not determined by this test.
Let   1
1 1 0... 0n n
nn nf x a x ax ax aa 
     
be a polynomial function.

Case 1
n Even
an > 0

Case 2
n Even
an < 0

Case 3
n Odd
an > 0

Case 4
n Odd
an < 0

EXAMPLE 3 Using the Leading-Term Test
Use the leading-term test to determine the end
behavior of the graph of
y = f x( ) = -2x3
+ 3x2
+ 4.
Solution
Here n = 3 (odd) and an = –2 < 0. Thus, Case 4
applies. The graph of f (x) rises to the left and
falls to the right. This behavior is described as
y ∞ as x –∞ and y –∞ as x ∞.

REAL ZEROS OF POLYNOMIAL FUNCTIONS
1. c is a zero of f .
2. c is a solution (or root) of the equation f
(x) = 0.
3. c is an x-intercept of the graph of f . The
point (c, 0) is on the graph of f .
If f is a polynomial function and c is a real
number, then the following statements are
equivalent.

EXAMPLE 4 Finding the Zeros of a Polynomial Function
Find all zeros of each polynomial function.
 
 
3 2
3 2
a. 2 2
b. 2 2
f x x x x
g x x x x
   
   
Solution
Factor f (x) and then solve f (x) = 0.

REAL ZEROS OF POLYNOMIAL FUNCTIONS
A polynomial function of degree n with real
coefficients has, at most, n real zeros.

EXAMPLE 6 Finding the Number of Real Zeros
Find the number of distinct real zeros of the
following polynomial functions of degree 3.
Solution
     
          
22
a. 1 2 3
b. 1 1 c. 3 1
f x x x x
g x x x h x x x
   
     

MULTIPLICITY OF A ZERO
If c is a zero of a polynomial function f (x)
and the corresponding factor (x – c) occurs
exactly m times when f (x) is factored, then c
is called a zero of multiplicity m.
m Behavior of f at x=c
Odd Crosses
Even touches

ODD MULTIPLICITY OF A ZERO

EVEN MULTIPLICITY OF A ZERO

EXAMPLE 7 Finding the Zeros and Their Multiplicity
Find the zeros of the polynomial function
f (x) = x2(x + 1)(x – 2), and give the multiplicity
of each zero.
Solution

TURNING POINTS
A local (or relative) maximum value of f is
higher than any nearby point on the graph.
A local (or relative) minimum value of f is lower
than any nearby point on the graph.
The graph points corresponding to the local
maximum and local minimum values are called
turning points. At each turning point the graph
changes from increasing to decreasing or vice
versa.

TURNING POINTS
The graph of f
has turning points
at (–1, 12) and at
(2, –15).
f x( ) = 2x3
- 3x2
-12x + 5

NUMBER OF TURNING POINTS
If f (x) is a polynomial of degree n, then the
graph of f has, at most, (n – 1) turning points.

EXAMPLE 8 Finding the Number of Turning Points
Use a graphing calculator and the window
–10  x  10; –30  y  30 to find the number of
turning points of the graph of each polynomial.
 
 
 
4 2
3 2
3 2
a. 7 18
b. 12
c. 3 3 1
f x x x
g x x x x
h x x x x
  
  
   

Solution
f has three total turning points; two local
minimum and one local maximum.
a. f x( ) = x4
- 7x2
-18

Solution continued
g has two total turning points; one local
maximum and one local minimum.
b. g x( ) = x3
+ x2
-12x

Solution continued
h has no turning points, it is increasing on the
interval (–∞, ∞).
c. h x( ) = x3
- 3x2
+ 3x -1

GRAPHING A POLYNOMIAL FUNCTION
Step 1 Determine the end behavior. Apply
the leading-term test.
Step 2 Find the zeros of the polynomial
function. Set f (x) = 0 and solve. The
zeros give the x-intercepts.
Step 3 Find the y-intercept by computing
f (0).

Step 4 Draw the graph. Use the multiplicities
of each zero to decide whether the
graph crosses the x-axis.
Use the fact that the number of turning
points is less than the degree of the
polynomial to check whether the graph
is drawn correctly.

EXAMPLE 9 Graphing a Polynomial Function
Sketch the graph of   3 2
4 4 16.f x x x x    
Solution
Step 1 Determine end
behavior.
Degree = 3
Leading coefficient = –1
End behavior shown in
sketch.

Solution continued
Step 2 Find the zeros by setting f (x) = 0.
Each zero has
multiplicity 1, the
graph crosses the
x-axis at each
zero.

Solution continued
Step 3 Find the y-intercept by computing f (0).
The y-intercept is f (0) = 16. The graph passes
through (0, 16).

Solution continued
Step 4 Draw the
graph.
The number of
turning points is 2,
which is less than
3, the degree of f.

Lecture 8 section 3.2 polynomial equations

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