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4.1 The Product Rule and Power
Rules for Exponents
• Review: PEMDAS (order of operations) –
note that exponentiation is number 2.
• Product rule for exponents:
a a
n
• Example:
m
5 5  5
2
3
a
23
nm
5
5
4.1 The Product Rule and Power
Rules for Exponents
• Power Rule (a) for exponents:
a 
m n
 a nm
• Power Rule (b) for exponents:
ab 
m
a b
m
• Power Rule (c) for exponents:
a
 
b
m
m
a
 m
b
m
4.1 The Product Rule and Power
Rules for Exponents
• A few tricky ones:
 2   2  2  2  8
3
3
 2  2   2  2  2  8
4
 2   2  2  2  2  16
4
4
 2  2   2  2  2  2  16
3
4.1 The Product Rule and Power
Rules for Exponents
• Examples (true or false):
t t  t
4
3
12
( t 4 ) 3  t 12
s  t 
2
s  t 
3
 s t
3
3
 s2  t 2
4.2 Integer Exponents and the
Quotient Rule
• Definition of a zero exponent:
a 0  1 (no matter wha t a is)
• Definition of a negative exponent:
a
n
1
1
 n  
a
a
n
4.2 Integer Exponents and the
Quotient Rule
• Changing from negative to positive
exponents:
m
n
a
b
 m
n
b
a
• Quotient rule for exponents:
m
a
mn

a
n
a
4.2 Integer Exponents and the
Quotient Rule
• Examples:
  10   1
0
1  2
0
3
x 2 x 2
 2
2
y
y
23
5
2
2
2
4.3 An Application of Exponents:
Scientific Notation
•
Writing a number in scientific notation:
1. Move the decimal point to the right of the first nonzero digit.
2. Count the places you moved the decimal point.
3. The number of places that you counted in step 2 is the
exponent (without the sign)
4. If your original number (without the sign) was
smaller than 1, the exponent is negative. If it was
bigger than 1, the exponent is positive
4.3 An Application of Exponents:
Scientific Notation
• Converting to scientific notation (examples):
6200000  6.2 10?
.00012  1.2 10?
• Converting back – just undo the process:
6.203 1023  620,300,000,000,000,000,000,000
1.86 105  186,000
4.3 An Application of Exponents:
Scientific Notation
• Multiplication with scientific notation:
4 10  5 10   4  5 10
8
5
5
108 
 20 103  2 101 103  2 10 2
• Division with scientific notation:
4 10   4  10
5 10  5 10
12
12
4
4

.8 1012 4  .8 108  8 10 7
4.4 Adding and Subtracting Polynomials;
Graphing Simple Polynomials
• When you read a sentence, it split up into
words. There is a space between each word.
• Likewise, a mathematical expression is split
up into terms by the +/- sign:
3x  4 x 2  3xy2  35
• A term is a number, a variable,
or a product or quotient of numbers and
variables raised to powers.
4.4 Adding and Subtracting Polynomials;
Graphing Simple Polynomials
• Like terms – terms that have exactly the
same variables with exactly the same
exponents are like terms:
5a b and  3a b
3 2
3 2
• To add or subtract polynomials, add or
subtract the like terms.
4.4 Adding and Subtracting Polynomials;
Graphing Simple Polynomials
• Degree of a term – sum of the exponents on
the variables
3 2
5a b degree  3  2  5
• Degree of a polynomial – highest degree of
any non-zero term
5x  3x  2 x  100 degree  3
3
2
4.4 Adding and Subtracting Polynomials;
Graphing Simple Polynomials
• Monomial – polynomial with one term
5x
3
• Binomial - polynomial with two terms
5y  y
2
• Trinomial – polynomial with three terms
5 x  3x  100
3
2
• Polynomial in x – a term or sum of terms of
n
4
2
the form ax for example : x  3x  x
4.5 Multiplication of Polynomials
• Multiplying a monomial and a polynomial:
use the distributive property to find each
product.
Example:
2
4 x 3 x  5
 4 x 3 x   4 x 5
2
 12 x 3  20 x 2
2
4.5 Multiplication of Polynomials
• Multiplying two polynomials:
x2
x3
 3x  6
x2  2x
x  x6
2
4.5 Multiplication of Polynomials
•
Multiplying binomials using FOIL (First –
Inner – Outer - Last):
1.
2.
3.
4.
5.
F – multiply the first 2 terms
O – multiply the outer 2 terms
I – multiply the inner 2 terms
L – multiply the last 2 terms
Combine like terms
4.6 Special Products
• Squaring binomials:
x  y   x  2 xy  y
2
x  y   x 2  2 xy  y 2
2
2
2
• Examples:
2
2
2
2
m  3  m  23m  3  m  6m  9
5 z  1
2
 5 z   25 z   12  25 z 2  10 z  1
2
4.6 Special Products
• Product of the sum and difference of 2
terms:
x  y   x  y   x 2  y 2
• Example:
3  w 3  w  32  w2  9  w2
4.7 Division of Polynomials
• Dividing a polynomial by a monomial:
divide each term by the monomial
x3  5x 2 x3 5x 2
 2  2  x5
2
x
x
x
4.7 Division of Polynomials
• Dividing a polynomial by a polynomial:
2x2  x  2
2 x  1 4 x3  4 x 2  5x  8
4 x3  2 x 2
 2 x 2  5x
 2x2  x
4x  8
4x  2
6