(Section 0.6: Polynomial, Rational, and Algebraic Expressions)
0.6.1
SECTION 0.6: POLYNOMIAL, RATIONAL, AND
ALGEBRAIC EXPRESSIONS
LEARNING OBJECTIVES
• Be able to identify polynomial, rational, and algebraic expressions.
• Understand terminology and notation for polynomials.
PART A: DISCUSSION
• In Chapters 1 and 2, we will discuss polynomial, rational, and algebraic
functions, as well as their graphs.
PART B: POLYNOMIALS
Let
n
be a nonnegative integer.
An
n
th
-degree polynomial in
x
, written in descending powers
of
x
, has the
following general form
:
a
n
x
n
+
a
n
±
1
x
n
±
1
+
...
+
a
1
x
+
a
0
,
a
n
²
0
()
The coefficients
, denoted by
a
1
,
a
2
,
…
,
a
n
, are typically assumed to be real
numbers, though some theorems will require integers or rational numbers.
a
n
, the leading coefficient
, must be
nonzero
, although any of the other
coefficients could be zero (i.e., their corresponding terms could be “missing”).
a
n
x
n
is the leading term
.
a
0
is the constant term
. It can be thought of as
a
0
x
0
, where
x
0
=
1
.
• Because
n
is a nonnegative integer,
all
of the exponents on
x
indicated above
must be nonnegative integers, as well. Each exponent is the degree
of its
corresponding term.