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Polynomials
Building Blocks of Algebra
• Numbers:
– Natural Numbers: 0,1,2,3 ….
– Integers: … -3, -2, -1, 0, 1, 2, 3 …
– Rational numbers can be written as
a
b
where a and b are integers and b is not zero
3 -7 1 19 97
, , , ,
5 2 2 3 12
a
• Not all symbols
rational numbersb
represent different
Definition:
Equality of Rational Numbers
• Two expressions a and c represent
d
b
(or or are names for the same rational
number if
a d b c
In this case we write
a
b

c
d
Many families of numbers
Complex
Reals
= certain rigid motions of the plane,
represent points on a line
= numbers represented by all possible decimal
expansions, represent points on a line
Rationals
= ratios of whole numbers, fractions, numbers
with terminating or repeating decimal representations
Integers
= -3, -2, -1, 0, 1, 2, … “whole numbers”
Diagram showing relations
between the number types
Numerals: Names for Numbers
3 6 -39
, ,
2 4 -26
Are all different names for the same number. In some
cases one name is more useful than another.
3
4

75
100
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”)
Arithmetic Expressions are
Numerals
(2*3 + 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”.
3
4
14 1
1
1
(
2

4 2
)=
( -------- ) = …. =
4
2 14 2
1.414213562
1
5

-0.3291502622
7
2
5

1

  7 
5 
5
3 4 i
e
1
3
10
2.718281828
3.1415926535897932384626433832795028841971693993751
2
3
9 3 *10 5 *10  7 *10
5
3
10
1

7
10
2

0.3333333333333333
4
10
7539
5
5.35005
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)
Basic Assumptions about Numbers
• Axioms of Arithmetic
–
–
–
–
–
Two fundamental numbers: 0 and 1
Two rules of combination: “+” and “*”
Commutativity: a+ b = b+a, a*b = b*a
Associativity: a+(b+c) = (a+b)+c, a*(b*c) = (a*b)*c
Relationship of rules (distributive law): a*(b+c) = a*b+a*c
• 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 …..
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
2
3
3
x
x
y

then expressions such as :2x, (x+3y) ,
,
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
Polynomials: monomials
A monomial in the unknown x is an expression of the form
cx
j
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 (different from x)
If c is not zero then the degree (in x) is the power j 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
20 x
0
( 2 u w ) x
10
2 5
7x y
3x
2
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
3
13
4
2
4
4 x  7 x 12 x  x  12 x  5 x
Representation not unique:
2
2
2
x x2 x xx
Standard form(s): terms collected descending increasing)
powers of x
13
4
3
2
12 x  x  4 x  12 x  7 x
The above have coefficients in the set of integers.
Degree of a polynomial in x
The degree of a polynomial, f, in x is the largest
x-degree of a monomial in
3
3
2
degree of 5 x  2 x 3 x  x  2 x
3
is 2
2
x  2 x
Since in a standard form it is
The polynomial
2
3
x  4 x y  7 x y 5 =
=
2
3
x  ( 4 y  7 y ) x 5
3
2
4 x y  x  7 x y 5
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 ydegree is 3.
Basic Algebra of Polynomials:
Addition/subtraction
f( x ) a0 a1 x1 ... an x
g( x ) b0 b1 x1 ... bm x
n
m
n
m
f( x ) g( x ) a0 a1 x1 2 ... an x  b0 b1 x1 bm x
f( x ) 3 4 x 5 x
2
2
3
g( x ) 2 x x  x
2
2
3
f( x ) g( x ) 3 4 x 5 x  2 x x  x
2
3
f( x ) g( x ) 5 3 x 4 x  x
(Standard form)
Basic Algebra of Polynomials:
Multiplication
Multiplication of polynomials is nothing but the distributive law
2
3
f( x ) 4 3 x 5 x  7 x
g( x ) 2 6 x 4 x
2
2
3
2
f( x ) g( x ) ( 4 3 x 5 x  7 x ) ( 2 6 x 4 x )
2
f( x ) g( x ) f( x ) ( 2 6 x 4 x ) = 2 f( x ) 6 x f( x ) (  4 x 2 ) f( x )
= 2( 4 3 x 5 x 2 7 x 3 ) + 6x( 4 3 x 5 x 2 7 x 3 ) +
4 x
2
(
2
3
4 3 x 5 x  7 x
)
By the distributive law this is just the sum of the product of
each monomial of f(x) with each monomial of g(x)
Using polynomial multiplication
Polynomial operations have tons of
applications. Multiplication can be used to
answer some counting questions (‘How
many?’ questions)
How many ways to make change for a dollar
using half dollars and quarters?
Example
2
3
( 23 x x ) ( 5x x )
3
=
3
2
3
2 ( 5 x x ) 3 x ( 5 x x ) (  x ) ( 5 x x )
=
3
2
4
2
3
5
10 2 x 2 x  15 x 3 x  3 x  (  5 ) x  x  x
=
5
4
3
2
 x  3 x  3 x  8 x  13 x 10
(standard form)
Finding one Coefficient
In standard form
2
3
( 2 3 x x ) ( 5 x x )
=
3
... c x  ...
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
2* x
3
2
+  x *(-x)
=
3
3
3
2 x  x  3 x
So c = 3
Application
Problem: How many ways can you make
change for a dollar using quarters and half
dollars?
Solution: Think about it. The answer is the coefficient
Of the x^100 term of the product.
(x^0 + x^50 + x^100)*(x^0 + x^25 + x^50 + x^75 + x^100)
= ….. + __c__ x^100 + …..
c x^100 = x^0*x^100 + x^50*x^50 + x^100*x^0 = 3*x^100
so c = 3. Three ways to make change: 0 halfs 4 quarters,
1 half 2 quarters, 2 halfs 0 quarters
Another application
Problem: How many different ways can you
choose 2 colors from a set of 4 colors?
Solution: in the product, choosing or not choosing a
color is choosing either the term x^1 or x^0 from the
factor. So, the coefficient of the x^2 term records the
number of ways of choosing 2 colors from a set of 4.
(x^0 + x^1)(x^0 + x^1)(x^0 + x^1)(x^0+x^1) =
…. + c * x^2 + …..
We can work out that c = 6.
For Monday
Read chapter 1 on Polynomials. Start
Working on Thursdays homework.
Print off. Work. Submit. Rework. Resubmit.
Continue until you get at least
40%.