Chapter 3
Polynomials
Sections Covered:
3.1 Remainder and Factor Theorems
3.2 Analyzing Polynomial Graphs
3.3 Zeros of Polynomials
3.4 Fundamental theorem of Algebra
3.5 Graphs of Rational Functions
~ November 2013 ~
Sun
17
24
Mon
Tue
Wed
Thu
Fri
14 B
15 A
3.1 “The Remainder
and Factor Theorems”
Long Division
3.1 “The Remainder
and Factor Theorems”
Long Division
HW 3.1 Long Division
HW 3.1 Long Division
21 A
22 B
18 B
19 A
20 B
3.1 “The Remainder
and Factor Theorems”
Synthetic Division
3.1 “The Remainder
and Factor Theorems”
Synthetic Division
Quiz: Long Division
Quiz: Long Division
3.3 “Zeros of
3.2 “Analyzing Graphs 3.2 “Analyzing Graphs Polynomial Functions”
of Polynomials”
of Polynomials”
Rational Zeroes
HW 3.1 Synthetic
Division
HW 3.1 Synthetic
Division
HW 3.2
HW 3.2
HW 3.3 Rational Zeros
25 A
26 B
27 Thanksgiving
28 Thanksgiving
29 Thanksgiving
3.3 “Zeros of
Polynomial Functions”
Rational Zeroes
3.3 “Zeros of
Polynomial Functions”
Descarte’s Rule
Break
Break
Break
HW 3.3 Rational Zeros
HW 3.3 Descarte’s Rule
Sat
16
23
30
~ December 2013 ~
1
8
15
2A
3B
4A
5B
6A
3.3 “Zeros of
Polynomial Functions”
Descarte’s Rule
Quiz 3.1-3.3 Review
Quiz 3.1-3.3 Review
***Quiz: 3.1-3.3***
***Quiz: 3.1-3.3***
HW 3.3 Descarte’s Rule HW Review 3.1-3.3
HW Review 3.1-3.3
Pre-Lab Worksheet
Pre-Lab Worksheet
9B
10 A
11 B
12 A
13 B
3.4 “The Fundamental
Theorem of Algebra”
3.4 “The Fundamental
Theorem of Algebra”
Bungee Barbie LAB
Bungee Barbie LAB
3.5 “Graphs of Rational
Functions”
HW 3.4
HW 3.4
16 A
17 B
14
HW 3.5
18 A
19 B
3.5 “Graphs of Rational Chapter 3 Test Review Chapter 3 Test Review ***Chapter 3 Test***
Functions”
HW 3.5
7
HW Chapter 3 Test
Review Packet
20 A
21
***Chapter 3 Test***
HW Chapter 3 Test
Review Packet
Table of Contents
Notes 3.1: Long Division
pg. 3-4
Notes 3.3: Descartes Rule
pg. 22-24
HW 3.1: Long Division
pg. 5-6
HW 3.3: Descartes Rule
pg. 25-26
Notes 3.1: Synthetic Division
pg. 7-9
Notes 3.4: Fund. Thm of Algebra
pg. 27-28
HW 3.1: Synthetic Division
pg. 10-11
HW 3.4: Fund. Thm of Algebra
pg. 29-30
Notes 3.2: Analyzing Graphs
pg. 12-15
Notes 3.5: Graphs of Rat. Fun.
pg. 31-33
HW 3.2: Analyzing Graphs
pg. 16-17
HW 3.5: Graphs of Rat. Fun.
pg. 34-36
Notes 3.3: Rational Roots
pg. 18-19
HW Reflection Sheet
pg. 37-38
HW 3.3: Rational Roots
pg. 20-21
3.1 Long Division
We’ve learned how to multiply polynomials. Now, we’ll learn how to DIVIDE polynomials.
There are 2 ways to divide: long division and synthetic division. First, let’s go back to grade school and
review how to divide plain old regular numbers.
8 65180
Now, let’s apply the same process to polynomials.
EX 1] x  2 x 3  2 x 2  6 x  9
EX 2] 2 x  3 2 x 3  7 x 2  17 x  3
Check:
Check your solution by multiplying the
divisor by the quotient and adding the
remainder.
You Try:
1. (m2 – 7m – 11) ÷ (m – 8)
2. (a2 – 28) ÷ (a – 5)
3. (2x2 – 17x – 38) ÷ (2x + 3)
4. (n3 + 7n2 + 14n + 3) ÷ (n + 2)
5. (-5k2 + k3 + 8k + 4) ÷ (-1 + k)
3.1 Synthetic Division and the Remainder and Factor Theorems
Synthetic division works differently.
x 3  2x 2  6x  9
Divide
x – 2.
by
1. First, the divisor must be in the form: (x – k). In our example, k = 2.
2. If a term is missing, you MUST use a zero as a place holder.
3. Write the leading coefficient. Then multiply diagonally and add vertically,
multiply and add, etc. ***FILL IN BELOW****
4. The answer is interpreted as follows:
Work backwards.
The last number is the remainder.
The next number back is the constant.
The next number back is the x coefficient.
2
The next number back is the x coefficient.
3
The next number back is the x coefficient.
To divide
2
x 3  2x 2  6x  9
1
x
2
by
2
-6
-9
x
c
remainder 
And so on.
x – 2.
EX 3] Divide x 3  14 x  8 by ( x  4) using synthetic division.
Do not write two signs!
The Remainder Theorem:
If a polynomial P (x ) is divided by x  c  , then the remainder equals P (c ) .
TRANSLATION: Do synthetic division, the remainder is your answer.
In the example at the top of the previous page, divide x 3  2 x 2  6 x  9 by x – 2,
5
2
we got x + 4x + 2 +
.
x2
The Remainder Theorem states that f (2) will be equal to –5!
Example:
The Factor Theorem:
A polynomial function P (x ) has a factor of x  c  if and only if P (c)  0 .
That is, x  c  is a factor if and only if c is a zero of P.
Solutions = Roots = Zeros = x-Intercepts
When we find the roots of a polynomial equation, we are finding the places where the value of the function is
ZERO. If f (x) = 0, then x is a root!!!!
Examples:
If f(4) = 0
If f(-7) = 0
If f(3/2) = 0
then x = 4 is a zero
then x = -7 is a zero
then x = 3/2 is a zero
and (x – 4) is a factor.
and (x + 7) is a factor.
and (2x – 3) is a factor.
Watch what happens when we find f (3) for the function f (x) = x2 + 2x − 15
3
1 2 −15
_____________
Is your remainder 0? _______________
That means ________ is a root of the equation,
a zero of the function, and
an x-intercept on the graph!!
That also means that ________ is a factor of the polynomial.
And it means that ________ is also a factor, also know as
a reduced polynomial or
a depressed polynomial.
Getting zero in synthetic substitution is a big deal!!
EX 4] f (x) = x3 + x2 + 2x + 24. *******(NEED CALCULATOR)*******
Graph the function. (You will need to set your window.)
There is one real root at x   3 .
Because the degree of the polynomial is 3, we know there are two other roots.
They must be imaginary.
We will use synthetic substitution to divide out (x + 3). Then the quadratic formula will allow us to find the
other roots.
−3
1 1 2 24
_______________
So, using the coefficients of the quotient, we write
(x + 3)(x2 − 2x + 8) = 0
We already know x   3 . We use the quadratic formula
on the second ( ) and get
x 
b 
b 2  4ac
x 
=
=
2a
= 1  i 7.
=
Therefore, x   3 , 1  i 7 , 1  i 7
Do you remember “complex conjugates”?
Example Problems:
Use Synthetic Division:
1. (m2 – 7m – 11) ÷ (m – 8)
3. (2x2 – 17x – 38) ÷ (2x + 3)
2. (a2 – 28) ÷ (a – 5)
4. (n3 + 7n2 + 14n + 3) ÷ (n + 2)
5. (-5k2 + k3 + 8k + 4) ÷ (-1 + k)
Use the Remainder Theorem to find P(c).
6. P(x) = 2x3 – x2 + 3x – 1 , c = 3
Determine if
7. P(x) = 6x3 – x2 + 4x , c = -3
8. P(x) = -x3 + 3x2 + 5x + 30 , c = 8
3.2 Analyzing Graphs of Polynomials
3.3 Possible Rational Zeros
NOTE: P (x ) is a polynomial function.
ALL polynomial functions have graphs that are __________ _______________ __________ .
Zeros of function P (x ) : x – values for which p(x) = 0
Roots of equation P (x ) : x – values for which p(x) = 0
Multiple Zeros of a Polynomial Function:
If P (x ) has ( x  r ) as a factor exactly k times, then r is a zero of multiplicity k of P (x ) .
( x  r )k
muliplicity
Zero/Root
EX 1] Find the zeros of P (x ) and state the multiplicity of each zero.
P( x)  x  7 
2
x
 8
4
x
 5 2 x  1
3
_____ occurs as a zero of multiplicity _____ .
_____ occurs as a zero of multiplicity _____ .
_____ occurs as a zero of multiplicity _____ .
_____ occurs as a zero of multiplicity _____ .
y
f (x)
Even Root = Bounce Point
x
Odd Root = goes through
EX 2] Given P( x)  x  7 
2
x
 8
4
x
 5 2 x  1 .
3
a) Find the maximum number of roots. ________________
b) Which zero(s) create a bounce point? _________________
c) Which zero(s) does the graph go through? _________________
The Rational Zero Theorem:
WHY IS IT IMPORTANT: Narrows the search for rational zeros to a finite list.

If P( x)  a n x n  a n 1 x n 1    a1 x  a 0 has integer coefficients a n  0
p
and
is a rational zero (in lowest terms) of p, then
q
p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n .
EX 3] Find the roots of x 3  6 x 2  10 x  3  0 .
HINT: Apply the Rational Root Theorem to find the possible rational roots!
What is p? ________
What is q? ________
HINT: Use synthetic division on x 3  6 x 2  10 x  3  0 to locate a root!
(Use your calc to estimate a zero )
EX 4] Find the possible rational roots of 3x3  5 x 2  7 x  2  0 .
HINT: Apply the Rational Root Theorem to find the possible rational roots!
What is p? ________
What is q? ________
3.3 Descartes’ Rule of Signs
Descartes’ Rule of Signs:
WHY IS IT IMPORTANT: Narrows down even further the possible positive and
negative roots.
Let P(x) be a polynomial function with real coefficients and with terms arranged in order of
decreasing powers of x.


The number of positive real zeros of P(x) is equal to the number of variations in the sign of P(x), or
to that number decreased by an even integer.
The number of negative real zeros of P(x) is equal to the number of variations in sign of P(-x), or to
that number decreased by an even integer.
Ex 5] Use Descartes’ Rule of signs to determine both the number of possible positive and the number of
possible negative real zeros of each polynomial function.
a) P( x)  x 4  5x3  5x 2  5x  6
b) P( x)  2 x5  3x3  5x 2  8x  7
Number of variations: _________
Number of variations: _________
Number of possible positive real zeros: ________
Number of possible positive real zeros: ________
P(-x) =
P(-x) =
Number of variations: _________
Number of variations: _________
Number of possible negative real zeros: ________
Number of possible negative real zeros: ________
Steps for Finding the Zeros of a Polynomial Function with Integer Coefficients:
1) Gather General Information.
 Determine the degree n of the polynomial function.
 The number of distinct zeros of the polynomial function is at MOST n.

Apply Descartes’ Rule of Signs to determine the number of possible negative real zeros of each
polynomial.
2) Check rational zeros.
 Apply the Rational Zero Theorem to list rational numbers that are possible zeros.
 Use synthetic division to test the numbers in the list.
3) Work with the reduced/depressed polynomial.
 Each time a zero is found, obtain the reduced/depressed polynomial.
 Work to get a reduced polynomial of degree 2.
 Then, find its zeros by factoring or by applying the quadratic formula.
Quick Sketch
EX 6] Find the zeros of f ( x)  x 3  7 x 2  16 x  12 .
At most _________ zeros.
Rational Root Theorem – Possible rational zeros:
Descartes Rule:
Synthetic Division/Quad. Formula/Factoring
y
x
5
EX 7] Find the zeros of g ( x)  3x 4  23x3  56 x 2  52 x  16
Quick Sketch
y
x
5
3.4 The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra:
If P (x ) is a polynomial function of degree n  1 with complex coefficients,
then P (x ) has at least one complex zero.
The Linear Factor Theorem:
If P (x ) is a polynomial function of degree n  1 with leading coefficient an  0 ,
then P (x ) has exactly n linear factors.
P( x)  an x  c1 x  c2 x  c3 x  cn  where c1, c2, … , cn are complex numbers.
(real and/or imaginary)
The Number of Zeros of a Polynomial Function Theorem:
If P (x ) is a polynomial function of degree n  1 , then P (x ) has exactly n complex
zeros, provided each zero is counted according to its multiplicity.
EX 1] Find all zeros of the polynomial function and write the polynomial as a product of linear factors.
P( x)  x 4  6 x 3  10 x 2  2 x  15
The Conjugate Pair Theorem:
If a  bi  ( b  0 ) is a complex zero of a polynomial function with real coefficients,
then the conjugate a  bi  is also a complex zero of the polynomial function.
Example:
If  5i  3 is a zero, then __________ is also a zero.
If 8i is a zero, then __________ is also a zero.
EX 2] Find a polynomial function P (x ) that has the indicated zeros.
a) degree 3; 1, 2, and -3 as zeros
b) degree 3; real coefficients and zeros 2i and -3
c) degree 2; real coefficients and a zero is 3 – 7i
Practice: Find all the zeros for the equation x 4  5x 3  4 x 2  3x  9  0 .
Hint: The two complex zeros are: x 
1  i 3 1  i 3
,
2
2
3.5 Graphs of Rational Functions
Rational Function can be written in the form f ( x) 
p ( x)
where p (x ) and q(x) are
q ( x)
polynomials and q(x) is NOT the zero polynomial.
Note
1.
The domain of a rational function of x includes all real numbers except the
x-values that would make the denominator equal to zero.
The Graph of a Rational Function:
Removable Discontinuity (Hole in the graph)
 occurs when p(x) and q(x) have a common factor
2.
Non-removable Discontinuity (Vertical Asymptote)
 occurs when the denominator equals zero
3.
Horizontal Asymptote
 the value that the function approaches as x increases without bound
a.
If the degree of the numerator < the degree of the denominator;
 y = 0 (the x-axis) is the horizontal asymptote
b.
If the degree of the numerator = the degree of the denominator;
 y
c.
lead coefficien t of the numerator
is the H.A.
lead coefficien t of the denominato r
If the degree of the numerator > the degree of the denominator;
 there is NO horizontal asymptote
4.
x-intercept  zero(s) of the numerator
5.
y-intercept  the value of f( 0)
6.
Slant Asymptote
 occurs when the degree of the numerator is EXACTLY one more than
the degree of its denominator
x2  x
f ( x) 
ex:
x  1
**Use long division to find the equation of the slant asymptote.
The slant asymptote will always be linear! Do NOT include the remainder
1.
x2  9
Graph f ( x) 
x  3
Advanced Functions - 3.5 Notes
y
Hole in graph:
__________
V.A.
__________
H.A.
__________
x-intercept
__________
y-intercept
__________
slant asymptote
__________
domain
__________
x
range ___________
y
2.
Graph g ( x) 
3x  17 x  20
x 2  5x  4
2
Hole in graph:
__________
V.A.
__________
H.A.
__________
x-intercept
__________
y-intercept
__________
slant asymptote
__________
domain
__________
x
range ___________
3.
4.
Graph f ( x) 
3x  18
x  4 x 2  12 x
3
Hole in graph:
__________
V.A.
__________
H.A.
__________
x-intercept
__________
y-intercept
__________
slant asymptote
__________
domain
__________
Graph g ( x) 
y
x
range ___________
2x 2  x
x  1
Hole in graph:
__________
V.A.
__________
H.A.
__________
x-intercept
__________
y-intercept
__________
slant asymptote
__________
domain
__________
y
x
range ___________
Chap. 3 Homework Completion Sheet
Name:
_________________________ Block: _______
Prerequisite Skills:
1. Long Division
2. Factoring
3. Using Complex Numbers
4. Solving Using
Quadratic Formula
Date
Assignment
Score
(out of 20)
Notes
What am I confident about?
Specifically, what was difficult or confusing
Am I weak on a prerequisite skill?