Using Properties of Exponents

1
6.1
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.
6.1 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
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
12
6.2
Evaluating and Graphing Polynomial Functions
What you should learn
Goal
1
Evaluate a polynomial function
Goal
2
Graph a polynomial function.
6.2 Evaluating and Graphing Polynomial Functions
13
Polynomial- is a single term or sum of two or
more terms containing variables in the numerator
with whole number exponents.
or
or
or
14
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.
15
Monomials-polynomials with one term.
Example) 6 or 2x or
Binomials-polynomials with two terms
Example)
Trinomials-polynomials with three terms.
Example)
16
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.
17
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
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Classifying polynomials by degree
Constant,
Degree 0,
Monomial
Linear,
Degree 1,
Binomial
Quadratic,
Degree 2,
Trinomial
Cubic,
Degree 3,
Monomial
Quartic,
Degree 4,
Polynomial
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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)
20
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
21
Synthetic Substitution
22
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
23
Reflection on the Section
Which term of a polynomial function is most
important in determining the end behavior of the
function?
assignment
24
6.3
Adding, Subtracting , and Multiplying Polynomials
What you should learn
Add, subtract, and multiply polynomials
Goal
1
6.3 Adding, Subtracting, and Multiplying
25
Add or subtract as indicated
ex)
ex)
26
Add or subtract as indicated
ex)
ex)
27
Add or subtract as indicated (vertically)
ex)
ex)
()
28
Add or subtract as indicated (vertically)
ex)
ex)
(-)
(-)
29
Use a vertical format to find each product
ex)

30
Multiplying Monomials To multiply monomials,
multiply the coefficients and then multiply the
variables. Use the product rule for exponents to
multiply the variables Keep the variable and
add the exponents.
multiply the coefficients and multiply the
variables
ex)
31
ex)
ex)
ex)
32
Finding the product of the monomial and the
polynomial
ex)
ex)
ex)
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Finding the product when neither is a monomial
ex)
ex)
34
Multiply by using the rule for finding the
product of the sum and difference
The Product of the Sum and Difference of Two
Terms
ex)
35
Multiply by using the rule for the Square of a
Binomial.
The Product of the Sum of Two Terms
ex)
36
Multiply by using the rule for the Square of a
Binomial.
The Product of the Difference of Two Terms
ex)
37
Using the FOIL Method to Multiply Binomials
ex)
38
Find the Product
ex)
ex)
39
Find the Product
ex)
ex)
40
Find the Product
ex)
ex)
41
Find the Product
ex)
ex)
42
Reflection on the Section
How do you add or subtract two polynomials?
assignment
43
6.4
Factoring and Solving Polynomial Equations
What you should learn
Goal
1
Factor polynomial expressions
Use Factoring to solve polynomial expressions
Goal
2
6.4 Factoring and Solving Polynomial Equations
44
Factoring is the process of writing a polynomial
as the product of two or more polynomials.
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.)
45
Directions
Find three factorizations for each monomial.
1.)
2.)
3.)
46
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
47
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.
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Factor each polynomial using the GCF.
ex)
ex)
ex)
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Factor each polynomial using the Greatest Common
Binomial Factor.
ex)
ex)
ex)
50
Factor by Grouping
Ex 1)
Group into binomials
Factor-out GCF from each binomial
Factor-out GCF
Factored by Grouping
51
Factoring the Sum or Difference of 2
Cubes 1.)Factoring the Sum of Two Cubes 2.)
Factoring the Difference of 2 Cubes
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Sum
Example 1)
or
53
Example2)
or
54
Difference
Example 3)
or
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Definition of a Quadratic Equation
A quadratic equation in x is an equation that can
be written in the standard form
where a, b, and c are real numbers, with a 0.
A quadratic equation in x is also called a
second-degree polynomial equation in x.
/
56
The Zero-Product Principle
If the product of two algebraic expressions is
zero, then at least one of the factors is equal
to zero.
If AB 0, then A 0 or B 0.
57
example)
According to the principle, this product can be
equal to zero if either
or
5
5
2
2
x 5
x 2
The resulting two statements indicate that the
solutions are 5 and 2.
58
Solve a Quadratic Equation by Factoring
example)
Factor the Trinomial using the methods we know.
(2x )(x ) 0
-

1
4
or
1
1
- 4
- 4
2x 1
x - 4
x 1/2
The resulting two statements indicate that the
solutions are 1/2 and - 4.
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Solve a Quadratic Equation by Factoring
example)
Move all terms to one side with zero on the
other. Then factor.
(x )(x ) 0
-
-
3
3
The trinomial is a perfect square, so we only
need to solve once.
3
3
x 3
The resulting two statements indicate that the
solutions are 3.
60
Reflection on the Section
How can you use the zero product property to
solve polynomial equations of degree 3 or more?
assignment
61
6.5
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.
6.4 The Remainder and Factor Theorem
62
Divide using the long division
ex)
x
7
63
Divide using the long division with Missing Terms
ex)
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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
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3. Write the leading coefficient of the dividend
on the bottom row.
3
1 4 -5 5
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
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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
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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
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Synthetic Division To divide a polynomial by x - c
Example 1)
-1
1 4 -2
-3
-1
1
3
-5
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Synthetic Division To divide a polynomial by x - c
Example 2)
2
1 0 -5 7
-2
4
2
1
2
5
-1
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Factoring a Polynomial
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).
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Factoring a Polynomial
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).
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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
73
6.6
Finding Rational Zeros
What you should learn
Goal
1
Find the rational zeros of a polynomial.
6.6 Finding Rational Zeros
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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
6.6 Finding Rational Zeros
75
Example 1)
Find the Rational Zeros of
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
6.6 Finding Rational Zeros
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Use Synthetic Division to decide which of the
following are zeros of the function 1, -1, 2, -2
Example 1)
-2
1 7 -4 -28
28
-10
-2
1
5
-14
0
x -2, 2
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Find all the REAL Zeros of the function.
Example 1)
1
1 4 1 -6
5
6
1
1
5
6
0
x -2, -3, 1
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Find all the Real Zeros of the function.
Example 2)
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
79
-1
1 3 7 5
-2
-5
-1
1
2
5
0
x 2, -1
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Reflection on the Section
How can you use the graph of a polynomial
function to help determine its real roots?
assignment
81
6.7
Using 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.
6.7 Using the Fundamental Theorem of Algebra
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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
83
Find all the ZEROs of the polynomial function.
Example 2)
3
1 0 1 0 -12
NOT DONE YET
3
9
30
90
1
3
10
30
0
84
Decide whether the given x-value is a zero of the
function.
, x -5
Example 1)
So, Yes the given x-value is a zero of the
function.
85
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)
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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
87
6.8
Analyzing Graphs of Polynomial Functions
What you should learn
Goal
1
Analyze the graph of a polynomial function.
Plot x-intercepts Find the Turning Points
The y-coordinate of a turning point is a Local
Maximum if the point is higher than all nearby
points. The y-coordinate of a turning points
is a Local Minimum if the point is lower that all
nearby points.
6.8 Analyzing Graphs of Polynomial Functions
88
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
89
6.1
Using Properties of Exponents
What you should learn
Goal
1
ghghhhghjghjghghggghjg
Goal
2
hghjghjghjghjghjgjhb
6.1 Using Properties of Exponents
90
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
91
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