CSE
1400
Applied Discrete Mathematics
Polynomials
Department of Computer Sciences
College of Engineering
Florida Tech
Fall
2011
Polynomials
1
Constants
1
Linear Polynomials
2
Quadratic Polynomials
2
The Power Basis
3
The Binomial Theorem
4
Horner’s Rule for Evaluating Polynomials
4
Taylor Polynomials
5
Falling Factorial Powers
5
Rising Factorial Powers
7
Problems on Polynomials
8
Abstract
Polynomial functions are computationally primitive and form bases for
approximating more complicated functions.
Polynomials
Polynomials are important because
1
. They are easy to evaluate.
2
. They can approximate arbitrarily well many more complex func
tions.
If
f
is a continuous realvalued function
on
[
a
,
b
]
and if any
e
>
0 is given, then
there exists a polynomial
p
on
[
a
,
b
]
such that

f
(
x
)

p
(
x
)

<
e
for all
x
in
[
a
,
b
]
. In words, any continuous
function on a closed and bounded
interval can be uniformly approximated
on that interval by polynomials to any
degree of accuracy.
Low degree polynomials
p
(
x
)
are studied in
college
algebra.
College
Instances of polynomials include
p
(
x
) =
3
x
+
2
p
(
x
) =
x
2

x

1
p
(
x
) =
x
2

4
p
(
x
) =
x
3
+
x
2
+
x
+
1
algebra studies how to solve polynomial equations
p
(
x
) =
0.
and
The
zeros
of polynomials can be com
puted.
3
x
+
2
=
0
at
x
=

2/3
x
2

x

1
=
0
at
x
=
1
±
√
5
2
x
2

4
=
0
at
x
=
±
2
x
3
+
x
2
+
x
+
1
=
0
at
x
=

1,
±
i
students learn to graph polynomial equations
p
(
x
) =
y
in a
Cartesian
coordinate system.
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cse 1400 applied discrete mathematics
polynomials
2
Constants
C
onstants are polynomials
. Constants are polynomials of de
gree
0
. There are famous, important constants.
•
0
,
Zero
•
1
,
One
•
√
2
≈
1.414213, The square root of 2
•
ϕ
= (
1
+
√
5
)
/2
≈
1.618033, The golden ratio
•
π
≈
3.141592, pi
•
e
≈
2.718281,
Euler
’s or Napier’s constant
•
γ
≈
0.577215, The EulerMascheroni constant
Linear Polynomials
P
olynomials that grow by a constant increment are
called linear
. A linear polynomial
p
(
x
)
is written
The
zero
of the equation
p
(
x
) =
0 is
x
=
m

1
b
.
p
(
x
) =
mx
+
b
for some
slope
m
∈
R
,
m
6
=
0 and
y

intercept
b
∈
R
. These
functions
are called first
degree
polynomials.
Lines through the origin are interesting. They have the form
ax
+
by
=
0
where
(
a
6
=
0
)
∨
(
b
6
=
0
)
Onedimensional affine space establishes an
equivalence
between
ax
+
by
is the inner product of two
vectors
h
a
,
b
i
and
h
x
,
y
i
. When the
inner product is
0
, the vectors are
orthogonal
(perpendicular).
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 Fall '11
 Shoaff
 Math, Binomial Theorem, Polynomials, Binomial, Horner, discrete mathematics polynomials

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