Transcript Document

Unit 5 Seminar
Prof. Maggie Habeeb
Unit 5 Seminar Agenda
• Exponents
What They Are, What They Look Like, How
They Work
Properties
• Scientific Notation
What It Is, What It Looks Like, How It Works
Converting To and From
• Polynomials
Vocabulary
Notation
Arithmetic
Exponents
• Exponents tell us how
many times we are to
multiply a number
times ITSELF.
• The number we are
multiplying is called
the BASE.
base
exponent
53 
24 
(2)4 
24  1* 2 * 2 * 2 * 2  8
Exponents (the base)
53  5 * 5 * 5  125
53 
24  2 * 2 * 2 * 2  16
23 
(2)4  2 * 2 * 2 * 2  16
(2)3 
24  1* 2 * 2 * 2 * 2  16
23 
Exponent Properties
Any non-zero base raised to the 0
power is equal to ONE
• EXAMPLES
1
( )0  1
3
30  1
(
22 0
) 1
3
23 / 90  1
w0  1
30  1
(3)0  1
 30  1*1  1
a0 
(a)0 
 a0 
2a 0 
2a 0 
( 3) 0  1
(2 x 2 ) 0  1
Any base raised to the 1 power is
equal to that base
• Examples
11
23 1
3 
( )  ( ) 
3
9
(1.24)1 
w1 
1
(3)1 
(2 x 2 )1 
Product Rule for Exponents
• If you multiply exponential
expressions with the
SAME base, keep the
common base and add
the exponents.
x3 x 4 x 4 
OR

x3 y 4 x5 
OR

Power Rule for Exponents
• To raise an exponent
to an exponent, keep
the base and
multiply the
exponents.
( x3 )2 
OR

( x3 x 4 )2 
OR

(4 x ) 
4 2
OR

Power of a Product Rule
•
To raise a product to a power,
raise each factor of the product
to that power.
 Essentially we are
distributing the exponent
 When using the Exponent
Power Rules, keep in mind
you can NOT distribute an
exponent across ADDITION
or SUBTRACTION.
( x  2)2 
(x y ) 
3 4 2
(2 y ) 
3 4
Powers of a Quotient Rule
•
To raise a quotient to a power, raise
the numerator and the denominator
to that power.
 Essentially we are distributing
the exponent
 When using the Exponent Power
Rules, keep in mind you can NOT
distribute an exponent across
ADDITION or SUBTRACTION.
x2 2
(
) 
3
x2 2
(
) 
3
x3 2
( 4) 
y
2 y2 3
( 4) 
5x
Negative Exponents
•
To change an exponent from
negative to positive, the base and
exponent must move.
 If the negative exponent is in a
numerator, it moves to the
denominator and becomes
positive
 If the negative exponent is in a
denominator, it moves to the
numerator and becomes
positive.
•
•
A negative exponent does NOT
indicate a negative number.
Negative numbers DO NOT move
… negative exponents DO.
3
x

4
y
3
x

4
y
x 3

4
y
Negative Exponents
•
To change an exponent from
negative to positive, the base and
exponent must move.
 If the negative exponent is in a
numerator, it moves to the
denominator and becomes
positive
 If the negative exponent is in a
denominator, it moves to the
numerator and becomes
positive.
•
•
A negative exponent does NOT
indicate a negative number.
Negative numbers DO NOT move
… negative exponents DO.
x 3 2
( 4) 
y
(
x 3 2
) 
y4
2

x 3
Quotient Rule for Exponents
• If you divide exponential
expressions with the
SAME base, keep the
common base and
subtract the exponents.
x5

4
x
OR 
x3

5
x
OR 
x3 y 4

5
x
OR 
Simplifying Exponential
Expressions
• Expressions containing exponents are
considered simplified when
No powers are raised to powers
No negative exponents (except in scientific
notation)
All like bases are combined
POLYNOMIALS
• A polynomial is a single term or the sum of
terms in which all variables have wholenumber exponents. No variable appears
in a denominator.
POLYNOMIALS are made up of
TERMS
• Terms can be placed into three categories:
CONSTANTS (plain old numbers; letters that
stay the same, like Pi)
VARIABLES (letters of the alphabet that are
used in place of the unknown)
PRODUCT of constants and variables (or
variables and other variables)
If the polynomial has
One term - it’s a MONOMIAL
Two terms - it’s a BINOMIAL
Three terms - it’s a TRINOMIAL
Four or more terms - it’s a POLYNOMIAL,
sometimes clarified by naming exactly how
many terms there are; ex. a polynomial
with six terms
Certain situations cause an expression to
NOT be a polynomial.
There’s a variable in a denominator
There’s a variable under a radical
There’s a number or variable with a
fractional or negative exponent
EXAMPLES OF POLYNOMOALS
5x
5x  3
5x  2 x  5
2
3r  5tw  vt
4
2
POLYNOMIAL FUNCTIONS
If I said to you … given the expression, 5x2  2 x  5
Find the value of the expression when x = 2 …. That would
mean for you to substitute 2 in every where there was a
x.
POLYNOMIAL FUNCTIONS
• Evaluate
f ( x)  5x2  2x  5
for f(2).
POLYNOMIAL FUNCTIONS
• Evaluate
7
f ( x) 
2x  5
for f(-2).
POLYNOMIAL ARITHMETIC
• Polynomials can be added, subtracted,
multiplied and divided just like other math
expressions
POLYNOMIAL
Addition/Subtraction
• ADDING and SUBTRACTING: Like so many other math
entities, we can only add and subtract LIKE TERMS.
35  8
3oranges  5oranges  8oranges
3x  5 x  8 x
3 x 2  5 x 2  3 x 2
3oranges  5cars  8orange cars they are NOT LIKE terms
3x+5y  8xy they are NOT LIKE terms
3x 3 - 9x does NOT combine .... they are NOT LIKE terms
5 x 2  4 y 2 does NOT combine .... they are NOT LIKE terms
3xy3 - 8xy3 + 5x 3 y = -5xy3 + 5x 3 y
POLYNOMIAL
Addition/Subtraction
•
•
ADDING and SUBTRACTING: Like
so many other math entities, we
can only add and subtract LIKE
TERMS.
To ADD or SUBTRACT
polynomials:
 Clear grouping symbols
by using the distributive
property
 Combine like terms
POLYNOMIAL
Addition/Subtraction
•
•
ADDING and SUBTRACTING: Like
so many other math entities, we
can only add and subtract LIKE
TERMS.
To ADD or SUBTRACT
polynomials:
 Clear grouping symbols by
using the distributive
property
 Combine like terms
 (8x 2  15x  4)  2(3x 2  x  1)
POLYNOMIAL Multiplication
• To MULTIPLY polynomials:
 Clear grouping symbols by
using the distributive
property
 Multiply coefficients
 Multiply variables
(utilizing exponent
properties as
needed)
POLYNOMIAL Multiplication
• To MULTIPLY polynomials:
 Clear grouping symbols by
using the distributive
property
 Multiply coefficients
 Multiply variables (utilizing
exponent properties as
needed)
 Combine like terms
 2 x (3x  x  1)
2
2
POLYNOMIAL Multiplication
• To MULTIPLY polynomials:
 Clear grouping symbols by
using the distributive
property
 Multiply coefficients
 Multiply variables (utilizing
exponent properties as
needed)
 Combine like terms
 (4 x  3)(2 x  5 x  7)
2
POLYNOMIAL Multiplication
• EXAMPLE
2 x2  5x  7

4x  3
2 x2  5x  7

4x  3
»
-6x2+15x-21
»
»
8x3-20x2+28x
8x3-26x2+43x -21