Transcript Chapter 6 polynomials 2007

Chapter 6
Polynomials and
Polynomial Functions
Polynomial Functions

Exploring Polynomial Functions
– Examples
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Modeling Data with Polynomial
Functions
– Examples
Degree
Name of
Degree
Number of Name using
number of
Terms
0
Constant
1
Monomial
1
Linear
2
Binomial
2
Quadratic
3
Trinomial
3
Cubic
4
Polynomial of
degree 4
4
Quartic
n
Polynomial of
degree n
5
Quintic
terms
x
0
3
5
6
9
11
12
14
y
42
31
26
21
17
15
19
22
Polynomials and Linear
Factors

Standard Form
– Example
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Factored Form
– Examples
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Factors and Zeros
– Examples
Writing a polynomial in standard form
You must multiply:
(x + 1)(x+2)(x+3)
X3 + 6x2 + 11x + 6
2x3 + 10x2 + 12x
2
2x(x
+ 5x +6)
Factor Theorem
The expression x-a is a linear factor
of a polynomial if and only if the
value a is a zero of the related
polynomial function.
Factors and Zeros
ZEROS
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-3
-2
-1
0
1
2
3
FACTORS
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(x
(x
(x
(x
(x
(x
(x
–
–
–
–
–
–
–
(-3)) or (x + 3)
(-2)) or (x + 2)
(-1)) or (x + 1)
0) or x
1)
2)
3)
Dividing Polynomials
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Long Division
Synthetic Division
Long Division
The purpose of this type of division is to
use one factor to find another.
4
)
40
Just as 4 finds the 10
x - 1 ) x3 + 6x2 -6x - 1
The (x-1) finds the (x2 + 7x + 1)
Synthetic Division
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When dividing by x – a, use synthetic
division.
The Remainder Theorem
The Remainder Theorem
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When using Synthetic Division, the
remainder is the value of f(a).
This method is as good as “PLUGGING
IN”, but may be faster.
Solving Polynomial
Equations
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Solving by Graphing
Solving by Factoring
Solving by Graphing
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Set equation equal to 0, then
substitute y for 0. Look at the xintercepts. (Zeros)
Let the left side be y1and let the right
side be y2. (Very much like solving a
system of equations by graphing).
Look at the points of intersection.
Solving by Factoring
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Sum of two cubes
(a3 + b3) = (a + b)(a2 – ab + b2)
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Difference of two cubes
(a3 – b3) = (a – b)(a2 + ab + b2)
More on Factoring
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If a polynomial can be factored into
linear or quadratic factors, then it can
be solved using techniques learned
from earlier chapters.
Solving a polynomial of degrees higher
than 2 can be achieved by factoring.
Theorems about Roots
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Rational Root Theorem
Irrational Root Theorem
Imaginary Root Theorem
Rational Root Theorem
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What are Rational Roots?
P’s and Q’s ………. ;)
Using the calculator to speed up the
process.
…And the Rational Roots
are…..
p
q
P includes all of the factors of the constant.
Q includes all of the factors
of the leading coefficient.
f(x) = x3 – 13x - 12
The possible rational roots are:
p = 12
q=1
 12,6,4,3,2,1
Test the Possible Roots…
In this case all roots are real
and rational, but you need only
to find one rational root. This will
become clear later.
Since -1, -3, and 4 are the Roots,
(x + 1), (x + 3), and (x – 4)
are the factors.
Multiply to show that
(x+1)(x+3)(x-4) = x3 – 13x – 12
(x+1)(x2 – x – 12)
x3 – x2 – 12x +x2 – x – 12
x3 – 13x – 12
Irrational Root Theorem
If a  b is a root,
then a  b is too.
These are called CONJUGATES.
Imaginary Root Theorem
If a  bi is a root,
then a  bi is too.
These are called CONJUGATES.
The Fundamental
Theorem of Algebra
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If P(x) is a polynomial of degree n  1
with complex coefficients, then P(x) = 0
has at least one complex root.
A polynomial equation with degree n will
have exactly n roots; the related
polynomial function will have exactly n
zeros.
The Binomial Theorem
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Binomial Expansion and Pascal’s
Triangle
The Binomial Theorem
PASCAL’S TRIANGLE
1
1
1
4
1
3
1
2
6
1
3
1
1
4
1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1
8 28 56 70 56 28 8
1
1
9 36 84 126 126 84 36 9
1