Download Unit 4 Exponents and Polynomials

Elementary Algebra
Exam 4 Material
Exponential Expressions &
Polynomials
Exponential Expression
a
n
• An exponential expression is:
where
is called the base and n is
called the exponent
• An exponent applies only to what it is
immediately adjacent to (what it touches)
• Example:
2
3x Exponent applies only to x, not to 3
4
 m Exponent applies only to m, not to negative
3
2x  Exponent applies to (2x)
a
Meaning of Exponent
• The meaning of an exponent depends on
the type of number it is
• An exponent that is a natural number
(1, 2, 3,…) tells how many times to
multiply the base by itself
2
3x  3  x  x
• Examples:
 m  1 m  m  m  m
3
3
2x   2x 2x 2x  8x
4
In the next section we will learn the meaning of any integer exponent
Rules of Exponents
• Product Rule: When two exponential
expressions with the same base are
multiplied, the result is an exponential
expression with the same base having an
exponent equal to the sum of the two
exponents
m
n
m n
a a  a
• Examples:
4 2
3 3  3  3
11
7
4
7 4
x x  x  x
4
2
6
Rules of Exponents
• Power of a Power Rule: When an
exponential expression is raised to a
power, the result is an exponential
expression with the same base having an
exponent equal to the product of the two
exponents
m n
mn
• Examples:
a 
3 
x 
a
4 2
 3
7 4
 x
42
74
 3
28
 x
8
Rules of Exponents
• Power of a Product Rule: When a
product is raised to a power, the result is
the product of each factor raised to the
n
power
n n
• Examples:
ab 
a b
3x 
 3 x  9x
2 y 
 2 y  16 y
2
4
2
4
2
4
2
4
Rules of Exponents
• Power of a Quotient Rule: When a
quotient is raised to a power, the result is
the quotient of the numerator to the power
and the denominator to the power
n
• Example:
a
a
   n
b
b
2
2
3
3
   2 
x
x
n
9
2
x
Rules of Exponents
• Don’t Make Up Your Own Rules
• Many people try to make these rules:
a  b   a  b
n
n
n
a  b   a  b
n
• Proof:
n
n
NOT TRUE! !!!!
NOT TRUE! !!!!
3  2  3  2
2
2
2
3  2  3  2
2
2
2
Using Combinations of Rules to
Simplify Expression with Exponents
• Examples:
52m p
  5  2 m p 5 16m p  80m p
 5x y    5  x y  125x y
2 x y   3x y   8x y  9x y  72x y
2 x y   8x y  8 x
9y
 3x y  9 x y
2
2
3 4
2
3 3
3 3
2
5 2
8
12
3
2
3 3
2
4
6
3 2
9
4
10
12
6
9
6
6
8
9
4
2
6
8
12
9
10 15
Homework Problems
• Section:
• Page:
• Problems:
4.1
261
Odd: 5 – 11, 25 – 79
• MyMathLab Section 4.1 for practice
• MyMathLab Homework Quiz 4.1 is due for
a grade on the date of our next class
meeting
Integer Exponents
• Thus far we have discussed the meaning
of an exponent when it is a natural
(counting) number: 1, 2, 3, …
• An exponent of this type tells us how many
times to multiply the base by itself
• Next we will learn the meaning of zero and
negative integer exponents
0
• Examples:
5
2
3
Integer Exponents
• Before giving the definition of zero and
negative integer exponents, consider the
4
pattern: 2 4  16
3  81
3
2 8
33  27
32  9
22  4
1
1
3 3
2 2
0
0
1
3

2 1
1
1
1
1
1
1  
 
1
1

 
 
3

2 
2
22 
2  2
1 1
 
4 2
2
3 
3  3 2
1 1
 
9 3
Definition of Integer Exponents
• The patterns on the previous slide suggest
the following definitions: a 0  1
a
n
1
 
a
n
• These definitions work for any base,
that is not zero:
3
5  1
0
1
1
2   
8
2
3
a,
Quotient Rule for Exponential
Expressions
• When exponential expressions with the same base are divided, the
result is an exponential expression with the same base and an
exponent equal to the numerator exponent minus the denominator
exponent
am
mn

a
an
Examples:
54
47
3
5

5

57
.
x12
12 4
8

x

x
x4
“Slide Rule” for Exponential
Expressions
• When both the numerator and denominator of
a fraction are factored then any factor may
slide from the top to bottom, or vice versa, by
changing the sign on the exponent
Example: Use rule to slide all factors to other
part of the fraction:
a mb  n
cr d s
 m n
r s
c d
a b
• This rule applies to all types of exponents
• Often used to make all exponents positive
Simplify the Expression:
(Show answer with positive exponents)
 
2
16
8y y
8 y 6 y 2
8  21
8 y 8
 1 4 1  1 3  3 8  11
1 4 1
y
2 y y
2 y y
y y
2 y
3
Homework Problems
• Section:
• Page:
• Problems:
4.2
270
Odd: 1 – 51, 57 – 77
• MyMathLab Section 4.2 for practice
• MyMathLab Homework Quiz 4.2 is due for
a grade on the date of our next class
meeting
Scientific Notation
• A number is written in scientific notation when it
is in the form:
a 10 , where 1  a  10 and n is an integer
n
Examples:
3.2  105
1.5342 10 9
20
 6.98 10
• Note: When in scientific notation, a single nonzero digit precedes the decimal point
Converting from Normal Decimal
Notation to Scientific
Notation
n
a 10
• Given a decimal number:
– Move the decimal to the right of the first non-zero digit
to get the “a”
– Count the number of places the decimal was moved
• If it was moved to the right “n” places, use “-n” as the
exponent on 10
• If it was moved to the left “n” places, use “n” as the exponent
on 10
• Examples:
320,000
Move decimal 5 places left
3.2  105
0.0000000015342 Move decimal 9 places right 1.5342 10 9
.
 698,000,000,000,000,000,000 Move decimal 20 places left
 6.98 10 20
Converting from Scientific
Notation to Decimal Notation
• Given a number in scientific notation: a 10
n
– Move the decimal in “a” to the right “n” places,
if “n” is positive
– Move the decimal in “a” to the left “n” places,
if “n” is negative
• Examples:
3.2  105
Move decimal 5 places right
1.5342 10
 6.98 10
20
9
Move decimal 9 places left
320,000
0.0000000015342
.
Move decimal 20 places right
 698,000,000,000,000,000,000
Applications of Scientific
Notation
• Scientific notation is often used in situations where the
numbers involved are extremely large or extremely small
• In doing calculations involving multiplication and/or
division of numbers in scientific notation it is best to use
commutative and associative properties to rearrange and
regroup the factors so as to group the “a” factors and
powers of 10 separately and to use rules of exponents to
end up with an answer in scientific notation
• It is also common to round the answer to the least
number of decimals seen in any individual number
Example of Calculations
Involving Scientific Notation
• Perform the following calculations, round
the answer to the appropriate number of
places and in scientific notation
3.2 10  6.98 10   3.2 6.98  10 10  
1.53
10 
1.5310 
14.59869281 10 10 10   14.5986928110 
5
5
20
20
9
9
5
20
9
34
What do we need to do to put this in scientific notation?
1.4598692811035  1.5  1035
Homework Problems
• Section:
• Page:
• Problems:
4.3
278
Odd: 1 – 9, 13 – 49, 63 – 75
• MyMathLab Section 4.3 for practice
• MyMathLab Homework Quiz 4.3 is due for
a grade on the date of our next class
meeting
Review of Terminology of Algebra
• Constant – A specific number
Examples of constants: 3  6
4
5
• Variable – A letter or other symbol used to
represent a number whose value varies or
is unknown
n
Examples of variables: x
A
Review of Terminology of Algebra
• Expression – constants and/or variables combined with
one or more math operation symbols for addition,
subtraction, multiplication, division, exponents and roots
in a meaningful way
Examples of expressions:
23
5 x
10
4
n
y  9 w
2
• Only the first of these expressions can be simplified,
because we don’t know the numbers represented by the
variables
Review of Terminology of Algebra
• Term – an expression that involves only a single
constant, a single variable, or a product
(multiplication) of a constant and variables
Examples of terms:
2
5
2
x y
2
m  5 x x  y
3
When constants and variables are
3
2
• Note:
multiplied, or when two variables are multiplied,
it is common to omit the multiplication symbol
Previous example is commonly written:
2
m
 5x
2
3
x y
2
2 5
xy
3
Review of Terminology of Algebra
• Every term has a “coefficient”
• Coefficient – the constant factor of a term
– (If no constant is seen, it is assumed to be 1)
• What is the coefficient of each of the
following terms?
2
m
2
5
2
1
2
 5x
1
3
x y
2 5
xy
3
2
3
Terminology of Algebra
• Every term has a “degree”
• Degree – the sum of the exponents on the
variables in the term
– (constant terms always have degree 0)
• What is the degree of each of the following
terms?
2
m
0
 5x
1
3
x y
2
2
2
5
2 5
xy
3
6
Review of Like Terms
• Recall that a term is a constant, a
variable, or a product of a constant and
variables
• Like Terms: terms are called “like terms”
if they have exactly the same variables
with exactly the same exponents, but may
have different coefficients
• Example of Like Terms:
2
3x y and
 7x y
2
Review of Like Terms
3
• Given the term: .24 xy
• Which of the following are like terms to this
one?
 4xy
3
3
5x y
2
 2x y
2
3
1 3
xy
2
Adding and Subtracting Like Terms
• When “like terms” are added or
subtracted, the result is a like term and its
coefficient is the sum or difference of the
coefficients of the other terms
• Examples:
 2x  7 x  x  4 x
4 x  19 x y  6 xy  2 x  x y  6 x 2  20 x 2 y  6 xy
2
2
2
2
Polynomial
• Polynomial – a finite sum of terms
• Examples:
6 x  5 x  4 How many terms ? 3
2
Degree of first term ? 2
Coefficien t of second term? - 5
3x y  5 x y
2
4
6
How many terms ? 2
Degree of second term? 10
3
Coefficien t of first term ?
Special Names for Certain
Polynomials
Number of Terms
Special Name
One term:
 9x y
Two terms:
3x y  5 x y
Three terms:
6 x  5x  4
2
2
2
monomial
4
6
binomial
trinomial
Evaluating Polynomials
• To “evaluate” a polynomial is to replace
variables with parentheses containing
specific numbers and simplify
• Evaluate the polynomial for x  3, y  4:
4  2y  x
4  2   
4 89
2

 13
2
4  2 4   3
2
Adding and Subtracting
Polynomials
• To add or subtract polynomials
horizontally:
– Distribute to get rid of parentheses
– Combine like terms
• Example:
2x
2
 

 3x  1  x  x  3  3x  2
2
2 x 2  3x  1  x 2  x  3  3x  2
3x 2  5 x
Adding and Subtracting
Polynomials
• To add or subtract polynomials vertically:
– Line up like terms in vertical columns
– Add or subtract terms in each column
• Example:
2x
2
 
 3x  1  3x 2  2
2 x 2  3x  1

 3x 2
2
 x 2  3x  3


Homework Problems
• Section:
• Page:
• Problems:
4.4
289
Odd: 1 – 55, 59 – 69,
73 – 77
• MyMathLab Section 4.4 for practice
• MyMathLab Homework Quiz 4.4 is due for
a grade on the date of our next class
meeting
Multiplying Polynomials
• To multiply polynomials:
– Get rid of parentheses by multiplying every
term of the first by every term of the second
using the rules of exponents
– Combine like terms
• Examples:
x  32 x 2  5x  4 
2x  35x  4 
2 x 3  5 x 2  4 x  6 x 2  15 x  12  2 x 3  x 2  11x  12
10 x 2  8 x  15 x  12 
10 x 2  7 x  12
Multiplying Binomials by FOIL
• As seen by the last example, we already know
how to multiply binomials by the general rule
(every term of first by every term of the second)
• With binomials, this is sometimes called the
FOIL method:
–
–
–
–
First times First
Outside times Outside
Inside times Inside
Last times Last
2x  35x  4 
10 x 2  8 x  15 x  12 
F
O I
L
10 x 2  7 x  12
Homework Problems
• Section:
• Page:
• Problems:
4.5
297
Odd: 1 – 55, 61 – 83
• MyMathLab Section 4.5 for practice
• MyMathLab Homework Quiz 4.5 is due for
a grade on the date of our next class
meeting
Squaring a Binomial
• To square a binomial means to multiply it
by itself
2x  32  2x  32x  3 
2
4 x 2  6 x  6 x  9  4 x  12 x  9
• Although a binomial can be squared by
foiling it by itself, it is best to memorize a
shortcut for squaring a binomial:
a  b 2 
2x  32 
a 2  2ab  b 2
4 x 2  12 x  9
first 2  2(first)(s econd)  second 2
Finding Higher Powers of
Binomials
• To find powers of binomials higher than
the second we use the definition of
exponents and the rules already learned
• Example:
2x  33  2 x  32 2 x  3  4 x 2  12x  92 x  3 
8 x 3  12 x 2  24 x 2  36 x  18 x  27 
8 x 3  36 x 2  54 x  27
Conjugate Binomials
• Two binomials are called “conjugates” if
they are exactly the same except for the
sign in the middle
• Examples: What is the conjugate of the
given binomial?
2 x  3 Conjugate is :
2x  3
5 x  4 Conjugate is :
5x  4
Multiplying Conjugate Binomials
• Conjugate binomials can be multiplied by
foil:
2x  32x  3  4 x 2  6 x  6 x  9  4 x 2  9
• However, it is best to memorize a formula
for multiplying conjugate binomials:
a  ba  b  a 2  b2
2x  32x  3 
4x2  9
(first)(fi rst) and last last 
Homework Problems
• Section:
• Page:
• Problems:
4.6
303
Odd: 3 – 19, 25 – 53
• MyMathLab Section 4.6 for practice
• MyMathLab Homework Quiz 4.6 is due for
a grade on the date of our next class
meeting
Dividing a Polynomial by a
Monomial
• Write problem so that each term of the
polynomial is individually placed over the
monomial in “fraction form”
• Simplify each fraction by dividing out
common factors
8x y 12xy
3
3
2
2

 4 xy  2  2 xy
8x y 12 xy 4 xy 2



2 xy
2 xy 2 xy 2 xy
1
2
4x  6 y  2 
xy
Dividing a Polynomial by a
Polynomial
• First write each polynomial in
descending powers
• If a term of some power is missing, write
that term with a zero coefficient
• Complete the problem exactly like a long
division problem in basic math
Example
 2x  3x 150 x  4
3x  2x  0x 150 x  0x  4
2
3
3
2
2
2
12 x  158
3x  2  2
x 4
x 2  0 x  4 3 x 3  2 x 2  0 x  150
 ( 3x 3  0 x 2  12 x )
 2 x 2  12 x  150
 ( 2x2  0x  8 )
12x 158
Homework Problems
• Section:
• Page:
• Problems:
4.7
312
Odd: 7 – 31, 39 – 75
• MyMathLab Section 4.7 for practice
• MyMathLab Homework Quiz 4.7 is due for
a grade on the date of our next class
meeting