Chapter 11
Polynomial Functions
11.1 Polynomials and Polynomial
Functions
Chapter 11
Polynomial Functions
11.1
Polynomials and Polynomial Functions
A polynomial function is a function of the form
f(x) = an x nn + an – 1 x nn – 11 +· · ·+ a 1 x + aa00
Where ann  00 and the exponents are all whole numbers.
For this polynomial function, aan is the leading coefficient,
coefficient
n
aa00 is the constant
constant term,
term and n is the degree.
degree
A polynomial function is in standard form if its terms are
descending order
order of
of exponents
exponents from
from left
left to
to right.
right.
written in descending
Objective: Determine whether a number is a root or zero of a given
equation or function.
Objective: Determine whether a number is a root or zero of a given
equation or function.
Objective: Determine whether a number is a root or zero of a given
equation or function.
Objective: Determine whether one polynomial is a factor of another by
division.
Objective: Determine whether one polynomial is a factor of another by
division.
Objective: Determine whether one polynomial is a factor of another by
division.
HW #11.1
Pg 483-484 1-21 Odd, 22-31, 35-36
Chapter 11
Polynomial Functions
11.2 Factor and Remainder Theorems
P(10) is the remainder when P(x) is divided by x - 10.
P(10) = 73,120
P(-8) = -37, 292
Find P( -4)
Yes
No
Yes
We look for linear factors of the form x - r. Let us try x - 1.
We know that x - 1 is not a factor of P(x).
We try x + 1.
To solve the equation P(x) = 0, we use the principle of zero products.
P(x) = (x – 2)(x + 3)(x + 5)
x = 2 x = -3 x = -5
f ( x )  D( x )Q( x )  R
f ( x )  ( x  1)Q( x )  0
f ( 1)  ( 1  1)Q( 1)  0
( 1)7  a( 1)  2  0
a  3
f ( x )  x 7  3x  2
f (2)  27  3(2)  2  120
x 3  3x 2  bx  5  ( x  2 )Q( x )  1
23  3(22 )  b(2)  5  (2  2)Q(2)  1
b  12
4. Solve
-5 < x< 1 or 2 < x < 3
HW #11.2
Pg 488-489 1-15 Odd, 16-31
Chapter 11
11.3 Theorems about Roots
Carl Friedrich Gauss was one of
the great mathematicians of all
time. He contributed to many
branches of mathematics and
science, including non-Euclidean
geometry and curvature of
surfaces (later used in Einstein's
theory of relativity). In 1798, at
the age of 20, Gauss proved the
fundamental theorem of algebra.
If a factor (x - r) occurs k times, we say that r is a root of multiplicity k
Where in the ____
did that come from?
The polynomial has 5 linear factors and 5 roots. The root 2 occurs 3
times, however, so we say that the root 2 has a multiplicity of 3.
-7 Multiplicity 2
4 Multiplicity 2
1 Multiplicity 1
3 Multiplicity 1
3 Multiplicity 2
-1 Multiplicity 1
Degree 3  3 roots
x9
x  3  4i
x  3  4i
Complex Roots Occur
in Conjugate Pairs
Irrational Roots also come in Conjugate Pairs
Degree 6  6 roots
x  2  5i
x  2  5i
x  i
xi
x  1 3
x  1 3
7  2i and 3  7 5
Degree 4  4 roots
2i  -2i
1. Divide p(x) by a known root to reduce it to a polynomial of
lesser degree
2. Divide the result by a different known root to reduce the degree
again
3. Repeat Steps 1 and 2 until you have reduced it to degree 2,
then factor or use the quadratic formula to find the remaining
roots
Roots are 2i, -2i, 2, and 3.
i,  i,  2, 1
x  2  ( x  2) is a factor
x  1  ( x  1) is a factor
x  3i  ( x  3i) is a factor
Let an = 1.
The number an can be any
nonzero number.
We proceed as in Example 6, letting an = 1
x  0  x is a factor
x  1  ( x  1) is a factor
x  4  ( x  4) is a factor
Degree 5  5 roots
Multiplicity 3 means it
is a factor 3 times
f ) p( x)  x3  6 x 2  3x  10
g ) p( x)  x5  6 x 4  12 x3  8x 2
x  1  2  x  1  2 is a root
x  1  3i  x  1  3i is a root
h) p( x)  x 4  6 x3  11x 2  10 x  2
i ) p ( x)  x 3  2 x 2  4 x  8
HW #11.3
Pg 494-495 1-49 Odd, 59
3
4
No
No
p( x )  2 x( x  (3  4i ))( x  (3  4i ))
p( x )  2( x  1)( x  2)( x  ( 2  i ))( x  ( 2  i ))
i , i , 1  2 , 1  2
Chapter 11
11.4 Rational Roots
List the possible rational zeros.
p :  1, 2, 3, 4, 6, 12
p
q
q : 1
Test these zeros using synthetic division.
p
q
The roots of ƒ are -1, 3, and -4.
List the possible rational zeros.
q:
p:
p 1 1 2 2 3 3 6 6
:  , , , , , , , ,
q 1 3 1 3 1 3 1 3
p
1
2
: 1,  , 2,  , 3, 6
q
3
3
Test these zeros using synthetic division.
p
1
2
: 1,  , 2,  , 3, 6 Test these zeros using synthetic
q
3
3
division.
1
The roots of ƒ are -2, , and 3 .
3
x=1
x = -1
HW # 11.4
Pg 499-500
1-11Odd, 13-21, 23-27 Odd
Chapter 11
11-5 Descartes’ Rule of Signs
Theorem 11-8 Descartes’ Rule Of Signs Part #1
The number of positive real zeros of a polynomial P(x)
with real coefficients is
a. the same as the number of variations of the sign of P(x),
or
b. Less than the number of variations of sign of P(x) by a
positive even integer
1
2
starts Pos.
changes Neg.
changes Pos.
f  x   x  x  3x  x  2
4
3
2
There are 2 sign changes so this means there could be
2 or 0 positive real zeros to the polynomial.
EXAMPLES
Determine the number of positive real zeros of the function
5
2
p
(
x
)

2
x

5
x
 3x  6
1
+
-
+
+
2 Sign Changes  2 or 0 Positive Real Roots
4
3
2
p
(
x
)

5
x

3
x

7
x
 12 x  4
2
+
-
+
-
+
4 Sign Changes  4, 2, or 0 Positive Real Roots
EXAMPLES
Determine the number of positive real zeros of the function
5
p
(
x
)

6
x
 2x  5
3
+
-
-
1 Sign Changes  Exactly 1 Positive Real Roots
Try This Determine the number of positive real zeros of the function.
a ) p ( x)  5 x 3  4 x  5
b) p( x)  6 x6  5x 4  3x3  7 x 2  x  2
c) p ( x)  3 x 2  2 x  4
Theorem 11-8 Descartes’ Rule Of Signs Part #2
The number of negative real zeros of a polynomial P(x)
with real coefficients is
a. the same as the number of variations of the sign of P(-x), or
b. Less than the number of variations of sign of P(-x) by a
positive even integer
1
starts Pos. changes Neg.
2
changes Pos.
f  x   x  x  3 x  x  2
4
3
2
There are 2 sign changes so this means there could be
2 or 0 negative real zeros to the polynomial.
EXAMPLES
Determine the number of negative real zeros of the function
4
3
2
p
(
x
)

5
x

3
x

7
x
 12 x  4
4
p( x)  5( x)4  3( x)3  7( x) 2  12( x)  4
p( x)  5 x 4  3x3  7 x 2  12 x  4
+
-
+
-
+
4 Sign Changes  4, 2, or 0 Negative Real Roots
Try This Determine the number of negative real zeros of the function.
d ) p ( x)  5 x 3  4 x  5
e) p( x)  6 x6  5x 4  3x3  7 x 2  x  2
f ) p ( x)  3x 2  2 x  4
68
67
69
If a sixth-degree polynomial with real coefficients has
exactly five distinct real roots, what can be said of one of its
roots?
Is it possible for a cubic function to have more than three real
zeros?
Is it possible for a cubic function with real coefficients to have no
real zeros?
HW #11.5
Pg 503 1-32
Chapter 11
11-6 Graphs of Polynomial Functions
3.
4.
5.
First, plot the x-intercepts.
Second, use a sign chart to
determine when f(x) > 0
and f(x) < 0
3
-1
+
+ 0 +
+ + 0 +
+
+
+
f(0) =3, Sketch a smooth
curve



First, plot the x-intercepts.
Second, use a sign chart to
determine when f(x) > 0
and f(x) < 0
1
-2
+ 0 +
+
- 0 +
+
-
+
+
f(0) =2, Sketch a Smooth
Curve



First, plot the x-intercepts.
Second, use a sign chart to
determine when f(x) > 0
and f(x) < 0
-1
-2
+ 0 +
+
+


3
+ +
- 0 +
0 + +
-
+
f(0) =-12, Sketch a Smooth
(0, -12)

A
B
3 x-intercepts
3 real roots.
1 x-intercept, 1 real root
2 x-intercepts,
2 real roots.
The left and right ends of a graph of an odd-degree
function go in opposite directions.
4 x-intercepts
4 real roots.
1 x-intercept,
1 real root
2 x-intercepts,
2 real roots.
3 x-intercepts,
3 real roots.
The left and right ends of a graph of an even-degree
function go in the same directions.
Even
Multiplicity
Odd
Multiplicity
3. Factor and make a sign chart.
5. Plot this information and consider the sign chart.
HW #11.6
Pg 507-508 1-22
Test Review
12
4. Solve
-5 < x< 1 or 2 < x < 3
The coefficient of xn-1 is the negative of the sum of the
zeros.
HW #R-11a
Pg 511-512 1-22
•
•
•
•
•
Prove the Remainder Theorem
Pg 489 #31
Pg 489 #32
Pg 503 #28
Find all the roots of a polynomial and use
them to sketch the graph
• Find roots on your calculator
• 2 parts
– No Calculator
– Calculator
• 1 Day Test
The graph of P  x   3x 4  2 x  12
can cross the x-axis in no more than r points. What is the value of r?
Use the rational root theorem to prove that the
7
2
is irrational by considering the polynomial p( x )  x  7
For what value of k will the remainder be the same when x 2  kx  4
is divided by x  1 or x  1
The equation x 2  2ax  b  0 has a root of multiplicity 2. Find it.
HW #R-11b
Pg 513 1-16