Using Properties of Exponents

1
5.1
Using Properties of Exponents
What you should learn
Goal
1
Use properties of exponents to evaluate and
simplify expressions involving powers.
Goal
2
Use exponents and scientific notation to solve
real-life problems.
L3.2.1
5.1 Using Properties of Exponents
2
Product of Powers Property
ex)
ex)
3
The Power of a Power Property
ex)
ex)
ex)
4
Power of a Product
ex)
ex)
5
Write each expression with positive exponents
only.
Negative Exponents in Numerators and
Denominators
and
ex)
ex)
6
Use the Zero-Exponent Rule
The Zero-Exponent Property
ex)
ex)
7
Divide by using the Quotient Rule
The Quotient of Powers Property
ex)
8
Simplify by using the Quotient of Powers Rule
The Power of Quotient Property
ex)
9
Simplify.
ex)
ex)
ex)
10
Simplify.
ex)
ex)
11
SCIENTIFIC NOTATION write the answer in
scientific notation.
Ex 1.)

X
Ex 2.)
X

12
Reflection on the Section
Give an example of a quadratic equation in vertex
form. What is the vertex of the graph of this
equation?
assignment
13
Evaluate and Graph Polynomial Functions
5.2
What you should learn
Goal
1

• Evaluate a polynomial function
• by using 2 kinds of Substitution
• Direct Substitution
• Synthetic Substitution

Goal
2
End Behavior of a functions graph.
L1.2.1
5.2 Evaluating and Graphing Polynomial Functions
14
Polynomial- is a single term or sum of two or
more terms containing variables in the numerator
with whole number exponents.
or
or
or
15
Polynomial- is a single term or sum of two or
more terms containing variables in the numerator
with whole number exponents.
It is customary to write the terms in the order
of descending powers of the variables. This is
Standard Form of a polynomial.
16
Monomials-polynomials with one term.
Example) 6 or 2x or
Binomials-polynomials with two terms
Example)
Trinomials-polynomials with three terms.
Example)
17
The Degree of If a does not equal zero, then
the degree of is n. The degree of a
nonzero constant is 0. The constant 0 has
no defined degree.
18
Polynomial
Degree of the number is the exponent of the
variable..
Example) 2x , has a degree of 1
Example) , has a degree of 2
Degree of the polynomial is the largest degree of
its terms.
Example)
, has a degree of 3
19
Classifying polynomials by degree
Degree Leading Coef Type
5,
Degree 0,
Constant
3,
Degree 1,
Linear
4,
Degree 2,
Quadratic
7,
Degree 3,
Cubic
1,
Degree 4,
Quartic
20
Polynomial function is a function of the form
where the exponents are whole numbers and
coefficients are real numbers.
Directions.
Decide whether the function is a polynomial
function. If so, write it in standard form and
stat its degree, type, and leading coefficient.
a)
b)
no
yes
c)
d)
no
yes
21
Goal
1
Evaluate a polynomial function
Directions Use Direct Substitution to evaluate
the Polynomial Function for
the given value of x.
, when x 3
f (x)
Make the Substitution.
f (3)
22
Another way to evaluate a polynomial function is
to use Synthetic Substitution.
Directions Use Synthetic Substitution to
evaluate the Polynomial
Function for the given value of x.
Synthetic Substitution
1. Arrange polynomials in descending powers, with
a 0 coefficient for any missing term.
NOTICE
23
Synthetic Substitution
24
Synthetic Substitution
Polynomial in standard form
x-value
2
0
-8
5
-7
3
add
6
18
30
105
multiply
2
6
10
35
98
25
Ch. 5.2 cont
END BEHAVIOR OF A FUNCTIONS GRAPH
Goal
2
Degree odd Leading Coefficient positive
Degree odd Leading Coefficient negative
as
as
5.2 Evaluating and Graphing Polynomial Functions
26
END BEHAVIOR OF A FUNCTIONS GRAPH
Degree even Leading Coefficient positive
Degree even Leading Coefficient negative
as
as
as
as
27
Directions
DECRIBE the degree and leading coefficient of the
polynomial function whose graph is
a)
b)
c)
ODD
EVEN
EVEN
Degree
Degree
Degree
NEG
POS
NEG
Leading Coef
Leading Coef
Leading Coef
28
DECRIBE THE END BEHAVIOR of the graph of the
polynomial function by completing these
statements
as
as
Ex 1)
29
Reflection on the Section
Which term of a polynomial function is most
important in determining the end behavior of the
function?
assignment
30
5.3
Add, Subtract , and Multiply Polynomials
What you should learn
Add, subtract, and multiply polynomials
Goal
1
Its just like Combining Like Terms.
A1.1.4
5.3 Adding, Subtracting, and Multiplying
31
Add or subtract as indicated
ex)
ex)
5.3 Adding, Subtracting, and Multiplying
32
Multiplying Monomials
multiply the coefficients and multiply the
variables
ex)
ex)
ex)
ex)
5.3 Adding, Subtracting, and Multiplying
33
Finding the product of the monomial and the
polynomial
ex)
ex)
ex)
5.3 Adding, Subtracting, and Multiplying
34
Finding the product when neither is a monomial
ex)
ex)
5.3 Adding, Subtracting, and Multiplying
35
Find the Product
ex)
5.3 Adding, Subtracting, and Multiplying
36
Reflection on the Section
How do you add or subtract two polynomials?
assignment
37
5.4
Factor and Solve Polynomial Equations
What you should learn
Goal
1
Factor polynomial expressions
New Factoring Methods -Difference of 2
Squares -Sum and Difference of 2 Cubes
-By Grouping
A1.2.5
5.4 Factoring and Solving Polynomial Equations
38
Factoring Monomials means finding two monomials
whose product gives the original monomial.
Or maybe
39
Factoring Monomials means finding two monomials
whose product gives the original monomial.
Can be factored in a few different ways
ex)
c.)
a.)
b.)
d.)
40
Directions
Find three factorizations for each monomial.
1.)
2.)
3.)
41
Find the greatest common factor.
1.)
and
GCF of 6 and 10 (or what divides into 6 and 10
evenly)
When dealing with the variables, you take the
variable with the smallest exponent as your GCF.
2.)
and
42
Factoring out the greatest common factor.
But, before we do thatdo you remember the
Distributive Property?
When factoring out the GCF, what we are going to
do is UN-Distribute.
43
Factor each polynomial using the GCF.
calculator
ex)
ex)
ex)
44
Factoring out the GCF and then factoring the
Difference of two Squares.
Example 1)
Whats the GCF?
5.2 Solving Quadratic Equations by Factoring
45
Factoring out the GCF and then factoring the
Difference of two Squares.
Example 2)
Whats the GCF?
5.2 Solving Quadratic Equations by Factoring
46
Factor by Grouping
Ex 1)
Group into binomials
Factor-out GCF from each binomial
Factor-out GCF
Factored by Grouping
47
Factor by Grouping
Ex 2)
Group into binomials
Factor-out GCF from each binomial
Factor-out GCF
Factored by Grouping
48
Sum
Example 1)
or
49
Factoring Perfect Square Trinomials
Example
Since both binomials are the same you can say
50
Reflection on the Section
How can you use the zero product property to
solve polynomial equations of degree 3 or more?
assignment
51
6.1
nth Roots and Rational Exponents
What you should learn
Evaluate nth roots of real numbers using both
radical notation and rational exponent notation
Goal
1
Goal
2
Evaluate the expression.
Goal
3
Solving Equations.
6.1 nth Roots and Rational Exponents
52
Using Rational Exponent
Notation Rewrite the expression using RATIONAL
EXPONENT notation.
Goal
1

• If n is odd, then a has one real nth root

Ex)

• If n is even and a gt 0, then a has two real nth
  roots

Ex)

• If n is even and a 0, then a has one nth root

• If n is even and a lt 0, then a has NO Real roots

6.1 nth Roots and Rational Exponents
53
Using Rational Exponent Notation Rewrite the
expression using RADICAL notation.
Ex)
Ex)
6.1 nth Roots and Rational Exponents
54
Evaluating Expressions Evaluate the expression.
Goal
2
Ex)
Ex)
6.1 nth Roots and Rational Exponents
55
Solving Equations
Goal
3
4
4
Ex)
Ex)
5
5
When the exponent is EVEN you must use the
Plus/Minus
When the exponent is ODD you dont use the
Plus/Minus
6.1 nth Roots and Rational Exponents
56
Solving Equations
4
4
Take the Root 1st.
Ex)
Very Important 2 answers !
6.1 nth Roots and Rational Exponents
57
Factoring Given expression
GCF
Factoring Bi-nomial
Factoring Tri-nomial
Diff of Squares
Quadratic Formula
Calculator Finding the Zeros
Into 2 Binomials ( )( ) Flaming
Banana
Diff of Cubes
Sum of Cubes
4 term Polynomial
Grouping
Finding the REAL-Number solutions of the equation.
58
5.5
Apply the Remainder and Factor Theorems
What you should learn
Goal
1
Divide polynomials and relate the result to the
remainder theorem and the factor theorem.

1. using Long Division
2. Synthetic Division

Goal
2
Factoring using the Synthetic Method
Goal
3
Finding the other ZEROs when given one of them.
A1.1.5
5.5 The Remainder and Factor Theorem
59
Divide using the long division
ex)
x
7
- ( )
- ( )
6.5 The Remainder and Factor Theorem
60
Divide using the long division with Missing Terms
ex)
- ( )
- ( )
- ( )
61
Synthetic Division To divide a polynomial by x - c
1. Arrange polynomials in descending powers, with
a 0 coefficient for any missing term.
2. Write c for the divisor, x c. To the
right, write the coefficients of the dividend.
3
1 4 -5 5
62
3
1 4 -5 5
3. Write the leading coefficient of the dividend
on the bottom row.
1
4. Multiply c (in this case, 3) times the value
just written on the bottom row. Write the
product in the next column in the 2nd row.
3
1 4 -5 5
3
1
63
5. Add the values in the new column, writing the
sum in the bottom row.
3
1 4 -5 5
3
add
1
7
6. Repeat this series of multiplications and
additions until all columns are filled in.
3
1 4 -5 5
21
3
add
16
7
1
64
7. Use the numbers in the last row to write the
quotient and remainder in fractional form. The
degree of the first term of the quotient is one
less than the degree of the first term of the
dividend. The final value in this row is the
remainder.
3
1 4 -5 5
48
3
21
add
1
7
16
53
65
Synthetic Division To divide a polynomial by x - c
Example 1)
-1
1 4 -2
-3
-1
1
3
-5
66
Synthetic Division To divide a polynomial by x - c
Example 2)
2
1 0 -5 7
-2
4
2
1
2
5
-1
67
Factoring a Polynomial
(x 3)
Example 1)
given that f(-3) 0.
2
11
18
9
-3
-6
-15
-9
2
5
3
0
multiply
Because f(-3) 0, you know that (x -(-3)) or (x
3) is a factor of f(x).
68
Factoring a Polynomial
(x - 2)
Example 2)
given that f(2) 0.
1
-2
-9
18
2
0
2
-18
1
0
-9
0
multiply
Because f(2) 0, you know that (x -(2)) or (x -
2) is a factor of f(x).
69
Reflection on the Section
If f(x) is a polynomial that has x a as a
factor, what do you know about the value of f(a)?
assignment
70
5.6
Finding Rational Zeros
What you should learn
Goal
1
Find the rational zeros of a polynomial.
L1.2.1
5.6 Finding Rational Zeros
71
The Rational Zero Theorem
Find the rational zeros of
solution
List the possible rational zeros. The leading
coefficient is 1 and the constant term is -12.
So, the possible rational zeros are
5.6 Finding Rational Zeros
72
Find the Rational Zeros of
Example 1)
solution
List the possible rational zeros. The leading
coefficient is 2 and the constant term is 30.
So, the possible rational zeros are
Notice that we dont write the same numbers twice
5.6 Finding Rational Zeros
73
Use Synthetic Division to decide which of the
following are zeros of the function 1, -1, 2, -2
Example 2)
-2
1 7 -4 -28
28
-10
-2
1
5
-14
0
x -2, 2
5.6 Finding Rational Zeros
74
Find all the REAL Zeros of the function.
Example 3)
1
1 4 1 -6
5
6
1
1
5
6
0
x -2, -3, 1
5.6 Finding Rational Zeros
75
Find all the Real Zeros of the function.
Example 4)
2
1 1 1 -9 -10
6
14
10
2
1
3
7
5
0
-1
1 3 7 5
-2
-5
-1
1
2
5
0
5.6 Finding Rational Zeros
76
-1
1 3 7 5
-2
-5
-1
1
2
5
0
x 2, -1
5.6 Finding Rational Zeros
77
Reflection on the Section
How can you use the graph of a polynomial
function to help determine its real roots?
assignment
5.6 Finding Rational Zeros
78
5.7
Apply the Fundamental Theorem of Algebra
What you should learn
Goal
1
Use the fundamental theorem of algebra to
determine the number of zeros of a polynomial
function.
THE FUNDEMENTAL THEOREM OF ALGEBRA
If f(x) is a polynomial of degree n where n gt
0, then the equation f(x) 0 has at least one
root in the set of complex numbers.
L2.1.6
5.7 Using the Fundamental Theorem of Algebra
79
Find all the ZEROs of the polynomial function.
Example 1)
-5
1 5 -9 -45
45
0
-5
1
0
-9
0
x -5, -3, 3
5.7 Using the Fundamental Theorem of Algebra
80
Decide whether the given x-value is a zero of the
function.
, x -5
Example 1)
-5
1 5 1 5
-5
0
-5
1
0
1
0
So, Yes the given x-value is a zero of the
function.
5.7 Using the Fundamental Theorem of Algebra
81
Write a polynomial function of least degree that
has real coefficients, the given zeros, and a
leading coefficient of 1.
-4, 1, 5
Example 1)
5.7 Using the Fundamental Theorem of Algebra
82
QUADRATIC FORMULA
83
Find ALL the ZEROs of the polynomial function.
Example )
x 2.732
x -.732
84
Find ALL the ZEROs of the polynomial function.
Example 24)
Doesnt FCTPOLYNow what?
85
Find ALL the ZEROs of the polynomial function.
Example )
86
Find ALL the ZEROs of the polynomial function.
Example )
-1
1 -4 4 10 -13 -14
-1
5
-9
-1
14
1
-5
9
1
-14
0
Graph this one.find one of the zeros..
87
Reflection on the Section
How can you tell from the factored form of a
polynomial function whether the function has a
repeated zero?
At least one of the factors will occur more than
once.
assignment