CHAPTER 4
Exponents and
Polynomials
Source:
Elementary & Intermediate Algebra, Fourth Edition
Prepared and Edited by:
Dr. Mohamad Hammoudi

Topics of Chapter 4
4.1: Positive Integer Exponents.
4.2: Zero and Negative Exponents and Scientific
Notation.
4.3: Introduction to Polynomials.
4.4: Addition and Subtraction of Polynomials.
4.5: Multiplication of Polynomials and Special
Products.
4.6: Division of Polynomials.
2

4.1: Positive Integer Exponents
Objectives:
4.1.1: Use exponential notation
4.1.2: Simplify expressions with positive integer
exponents
3

Exponents
Instead of writing 2 . 2 . 2 . 2 . 2 . 2 . 2, we may write
2
7
Instead of writing a . a . a . a . a, we may write a
5
We call (
a
) the
base
on the expression and (
5
) the
exponent
, or
power
.
a
5
is read as “
a to the fifth power.”
Definition of Exponential Expression:
An expression of this type is said to be in
exponential
form
.
4

4.1.1: Use Exponential Notation
Examples:
Write each of the following, using exponential
notation.
1.
3 . 3 . 3 . 3 . 3
2.
5y . 5y . 5y
3.
w . w . w . w
5

Product Rule for Exponents
Let’s consider what happens when we multiply two expressions
in exponential form with the
same base
.
Notice that the product is simply the base taken to the power
that is the
sum
of the two original exponents.
6

4.1.2: Simplify Expressions with Positive Integer Exponents
Examples:
Simplify each expression.
1.
b
4
. b
6
2.
(2a)
3
. (2a)
4
3.
(
–
2)
5
. (
–
2)
2
4.
(10
7
) . (10
11
)
7

4.1.2: Simplify Expressions with Positive Integer Exponents
Using the product rule for exponents together with the
commutative
and
associative
properties, simplify each
expression.
Multiply
the
coefficients
and
add
the
exponents
by the
product rule.
Examples:
Using the product rule for exponents together
with the commutative and associative properties, simplify
each expression.
1.
(x
4
) (x
2
) (x
3
) (x)
2.
(3x
4
) (5x
2
)
3.
(2x
5
y) (9x
3
y
4
)
4.
(
–
3x
2
y
2
) (
–
2x
4
y
3
)
8

Quotient
Rule for Exponents
9

4.1.2: Simplify Expressions with Positive Integer Exponents
Examples:
Simplify each expression.
1.
2.
3.
4.
5.
x
x
4
10
a
a
7
8
7w
63w
5
8
b
8a
b
32a
–
2
5
4
10
10
6
16
10

Product-Power Rule for Exponents
(xy)
3
= (xy)(xy)(xy)
= (x . x . x) . (y . y .y)
=
x
3
y
3
Examples:
Simplify each expression.
1.
(2x)
3
2.
(
–
4x)
4
11

Power Rule for Exponents
(3
2
)
3
= (3
2
)(3
2
)(3
2
)
= (3)(3)(3)(3)(3)(3)
= 3
6
Examples:
Simplify each expression.

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