Polynomials

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Polynomials
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Back to numbers

• Numbers
• Natural Numbers 0,1,2,3 .
• Integers -3, -2, -1, 0, 1, 2, 3
• Rational numbers can be written as
• where a and b are integers and b is not zero
• Not all symbols represent different rational
  numbers

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Definition Equality of Rational Numbers

• Two expressions and represent
• (or or are names for the same rational number
  if
• In this case we write

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Many families of numbers
Complex
certain rigid motions of the plane,
represent points on a line
Reals
numbers represented by all possible
decimalexpansions, represent points on a line
Rationals
ratios of whole numbers, fractions,
numberswith terminating or repeating decimal
representations
Integers
-3, -2, -1, 0, 1, 2, whole numbers
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Numerals Names for Numbers
Are all different names for the same number. In
some cases one name is more useful than
another. We often represent numbers with
denominator 100 in which case we give just the
numerator and say per hundred (more usually
per centrum or percent)
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Arithmetic Expressions are Numerals
(23 7) (6 7) 13 Each of these is a
numeral. The sign can be read is another
name for..

• We often decide to agree on a standard forms for
  numerals and define procedures for finding the
  standard form from a given one.Applying such
  procedures is called simplification.

14 -2
( ) ( -------- ) .
4
8
e
7539
5.35005
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• Much of algebra is looking for standard numerals
  (canonical forms)
• The rules of algebra allow us to
• transform one numeral for a number into another
  (arithmetic) ,
• compare numbers represented by numerals
  (order),
• know that some possible numerals actually
  represent numbers (completeness)

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Basic Assumptions about Numbers

• Axioms of Arithmetic
• Two fundamental numbers 0 and 1
• Two rules of combination and
• Commutativity a b ba, ab ba
• Associativity a(bc) (ab)c, a(bc)
  (ab)c
• Relationship of rules (distributive law)
  a(bc) abac
• Axioms of order (for real numbers)
• Every real number is (one of) positive, negative
  or zero
• The sum and product of positive numbers is
  positive
• Axiom of completeness (for real numbers)
• Every decimal is a numeral for some number.
• .123456789101112131415161718192021 ..

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Unknowns, variables, indeterminants, parameters

• Variables (e.g. x, y ,z ) are names (numerals)
  for unknown (or undisclosed) numbers. Although
  we may not know them exactly we do know that they
  are numbers and therefore they have the
  properties guaranteed by the axioms. If x and y
  are unknowns then expressions such as 2x, (x3y)
  , , etc. are also numbers.
• Parameters are thought of as variables whose
  values are taken from some specified set
• Ideterminants are symbols having only formal
  properties defined by the axioms
• For our purposes these terms can all be used
  interchangeably

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Polynomials monomials
A monomial in the unknown x is an expression of
the form
where j is a counting number and c represents a
number. The quantity c is a numeral which may,
itself, be an expression in terms of other
unknowns If c is not zero then the degree (in x)
is the power to which x appears in the
expression. If c 0 the monomial is 0. The 0
monomial does not have a degree. If it is a
monomial in more than one variable the degree is
the sum of the degrees in the individual variables
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Polynomials
A polynomial in the variable x with
coefficients in the set R is the sum of a set of
monomials in x where Each monomial has its
coefficients in R,
s
-
Representation not uniqueStandard form(s)
terms collected descending (increasing) powers of
x
The above have coefficients in the set of
integers.
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Degree of a polynomial in x
The degree of a polynomial, f, in x is the
largest x-degree of a monomial in
degree of
is 2
Since in a standard form it is
The polynomial


Is a polynomial in x with coefficients
polynomials in y and in y with coefficients
polynomials in x. Its x-degree is 2 and its
y-degree is 3.
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Basic Algebra of PolynomialsAddition/subtraction
(Standard form)
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Basic Algebra of PolynomialsMultiplication
Multiplication of polynomials is nothing but the
distributive law

2( ) 6x(
)
( )
By the distributive law this is just the sum of
the product of each monomial of f(x) with each
monomial of g(x)
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In how many ways can youwrite 6 as sum of two
numbersone number from each group?
1 51 51 51 53 33 35 15 1
A
B
0
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1 51 51 51 53 33 35 15 1
1 from A 5 from B in 2 2 ways
3 from A 3 from B in 1 2 ways
5 from A 1 from B in 1 2 ways
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Total for i j 6 (number of is in A)
(number of js in B)
Total for i from 0 to 6 (number of is in A)
(number of (6-i)s in B)
10 22 20 12 01 12 01 8
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Polynomials for bookkeepingGenerating
Polynomial for A
CoefficientNumber of ways to get the integer by
selecting oneelement of A
ExponentInteger
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Calculate coefficient of in
( 2 )(2 ) ( 1 )(2 )
(1 )( 2 ) (8 )
Calculate

For any monomial of the coefficient
is the number of ways the exponent can be the
result of adding one element of A to one element
of B
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Variation on the theme Suppose the set A are
values of some coins you have. Whatare the
amounts you can buy with exact change?
Answer the amounts are exactly theexponents of
the following product, the coefficients are the
number of ways it can be done.Exercise Write
down all 7 ways to make 8.


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How Many Ways are There to Make Change for a
Dollar?
P
Q
H
P
N
D
25


26

27

28

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This says that there are exactly 9 ways to make
change for 20 cents and exactly 50 ways to make
change for 50 cents.How many ways to make change
for a dollar?
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Example



(standard form)
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Finding one Coefficient
In standard form

Calculate c Solution The degree 3 monomial in
the product is the sum of all monomials of degree
3 that can be made by multiplying one monomial
from factor by a monomial from the other
So c 3


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Substitutions in Polynomials
Because the unknown in a polynomial can be any
number any number can be substituted for the
unknown and the result must be a number. If
f(x) is a polynomial and t is a number then we
denote by f(t) the result of substituting t for x
in f.
113 (standard form)

( standard form)
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Special Substitutions
If and a
is not zero, find a substitution x -gt xt so
that f(xt) has no x-term . That is the
coefficient of x in f(xt) is zero.
Solution f(xt)

In order for the coefficient of x to be zero we
must have b 2at 0 so
f(x5/6)
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Pascals Triangle
To simplify must calculate
X a, b , need
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The product is the sum of all monomials that can
be made by taking one term out of each factor.
More generally
The sum of all monomials of type
where there are n
terms,each taken from exactly one of the
factors.
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where the total
is the sum of all monomials
number of as and bs is n
If there are j as then there will be n j
bs so taking commutativity into account such a
term must be
This says, for instance
So to write down a standard form of
we just need to know the coefficients.
1
1 1
1 2 1
1 3 3 1
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Binomial Expansion(Pascals Triangle)
1
1 1
1 2 1
This says, for instance
1 3 3 1
1 4 6 4 1
1 __ __ __ __ _
Why does it work?
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Example
If what is
f(2x-3)?
By Pascals triangle
This is true for any a and b so it is true when a
2x and b 3

This plus (-4(2x-3) 7)
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Geometry Competing the Square
a
x
x
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Algebra Completing the Square
Problem Write
where
The geometric construction tells us how to do
this if it looks like
From the geometric completing square we have
Here a 5/2 so

Multiply through by 2
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Next

• Polynomials of degree 0 and 1
• Solving Linear Equations
• Reading Chapter 2, pp 15-23