Chapter 11Polynomial Functions
11.1 Polynomials and Polynomial Functions
Chapter 11Polynomial Functions
11.1
Polynomials and Polynomial Functions
A polynomial function is a function of the form
f (x) = an x n + an – 1 x
n – 1 +· · ·+ a 1 x + a 0
Where an 0 and the exponents are all whole numbers.
A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.
For this polynomial function, an is the leading coefficient,
a 0 is the constant term, and n is the degree.
an 0
an
an leading coefficient
a 0
a0 constant term n
n
degree
descending order of exponents from left to right.
n n – 1
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 #11.1Pg 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 Q x( ) ( 1) ( ) 0
f Q( 1) ( 1 1) ( 1) 0 a7( 1) ( 1) 2 0
a 3 f x x x 7( ) 3 2
f 7(2) 2 3(2) 2 120
3 23 5 2 1x x bx ( x )Q( x )
b Q 3 22 3(2 ) (2) 5 (2 2) (2) 1
b 12
4. Solve
-5 < x< 1 or 2 < x < 3
HW #11.2Pg 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
3 Multiplicity 1
4 Multiplicity 2
3 Multiplicity 2
1 Multiplicity 1
-1 Multiplicity 1
Degree 3 3 roots 3 4x i 9x 3 4x i
Complex Roots Occur in Conjugate Pairs
Irrational Roots also come in Conjugate Pairs
Degree 6 6 roots 2 5x i
x i
1 3x
2 5x i
x i
1 3x
7 2 3 7 5i and
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.
, , 2, 1i i
2 ( 2)x x is a factor 1 ( 1)x x is a factor 3 ( 3 )x i x i is a factor The number an can be any
nonzero number.
Let an = 1.
We proceed as in Example 6, letting an = 1 Degree 5 5 roots
0x x is a factor 1 ( 1)x x is a factor
4 ( 4)x x is a factor
Multiplicity 3 means it is a factor 3 times
3 2) ( ) 6 3 10f p x x x x
5 4 3 2) ( ) 6 12 8g p x x x x x
1 2 1 2x x is a root 1 3 1 3x i x i is a root
4 3 2) ( ) 6 11 10 2h p x x x x x
3 2) ( ) 2 4 8i p x x x x
HW #11.3Pg 494-495 1-49 Odd, 59
4 3
No
No
2 3 4 3 4 ( ) ( ( ))( ( ))p x x x i x i2 1 2 2 2 ( ) ( )( )( ( ))( ( ))p x x x x i x i
1 2 1 2, , ,i i
Chapter 11
11.4 Rational Roots
List the possible rational zeros.
: 1, 2, 3, 4, 6, 12p : 1q
pq
pq
Test these zeros using synthetic division.
The roots of ƒ are -1, 3, and -4.
List the possible rational zeros.
:p :q
1 1 2 2 3 3 6 6: , , , , , , , ,
1 3 1 3 1 3 1 3pq
1 2: 1, , 2, , 3, 6
3 3pq
Test these zeros using synthetic division.
1 2: 1, , 2, , 3, 6
3 3pq
Test these zeros using synthetic division.
The roots of ƒ are -2, , and .
13
3
x = 1
x = -1
HW # 11.4Pg 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 isa. the same as the number of variations of the sign
of P(x), orb. Less than the number of variations of sign of P(x)
by a positive even integer
23 234 xxxxxfstarts Pos. changes Neg. changes Pos.
1 2
There are 2 sign changes so this means there could be 2 or 0 positive real zeros to the polynomial.
Determine the number of positive real zeros of the function
EXAMPLES
15 2( ) 2 5 3 6p x x x x
+ - + +
2 Sign Changes 2 or 0 Positive Real Roots
24 3 2( ) 5 3 7 12 4p x x x x x
+ - + -
4 Sign Changes 4, 2, or 0 Positive Real Roots
+
Determine the number of positive real zeros of the function
EXAMPLES
35( ) 6 2 5p x x x
+ - -
1 Sign Changes Exactly 1 Positive Real Roots
Try This Determine the number of positive real zeros of the function.
3) ( ) 5 4 5a p x x x
6 4 3 2) ( ) 6 5 3 7 2b p x x x x x x
2) ( ) 3 2 4c p x x x
Theorem 11-8 Descartes’ Rule Of Signs Part #2
The number of negative real zeros of a polynomial P(x)
with real coefficients isa. the same as the number of variations of the sign of
P(-x), orb. Less than the number of variations of sign of P(-x)
by a positive even integer
There are 2 sign changes so this means there could be 2 or 0 negative real zeros to the polynomial.
23 234 xxxxxf
starts Pos. changes Neg. changes Pos.1 2
Determine the number of negative real zeros of the function
EXAMPLES
44 3 2( ) 5 3 7 12 4p x x x x x
+ - +
4 Sign Changes 4, 2, or 0 Negative Real Roots
4 3 2( ) 5( ) 3( ) 7( ) 12( ) 4p x x x x x
4 3 2( ) 5 3 7 12 4p x x x x x
- +
Try This Determine the number of negative real zeros of the function.
3) ( ) 5 4 5 d p x x x
6 4 3 2) ( ) 6 5 3 7 2 e p x x x x x x
2) ( ) 3 2 4 f p x x x
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 #11.5Pg 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
-1 3
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
-2 1
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
-2 -1
0
0+
--
+
-+
f(0) =-12, Sketch a Smooth Curve
+
--
3
(0, -12)
0
+
+
+
+ + - +
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.
The left and right ends of a graph of an even-degree function go in the same directions.
3 x-intercepts, 3 real roots.
Even Multiplicity
Odd Multiplicity
3. Factor and make a sign chart.
5. Plot this information and consider the sign chart.
HW #11.6Pg 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-11aPg 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 43 2 12P x x x
can cross the x-axis in no more than r points. What is the value of r?
2 7 ( )p x x
7Use the rational root theorem to prove that the
is irrational by considering the polynomial
2 4x kx 1x 1x
For what value of k will the remainder be the same when
is divided by or
The equation 2 2 0x ax b has a root of multiplicity 2. Find it.
HW #R-11bPg 513 1-16
