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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
Product of Powers Property
x x  x
m
ex)
x x
ex)
y y
7
n
9
x
4
y
7 9
1 4
mn
x
16
y
5
The Power of a Power Property
x 
m n
ex)
x 
6 4
x
x
x
64
43
ex)
(2 )
2
ex)
 3 
  3
4 3
7 5
mn
75
24
2
12
  3
35
Power of a Product
xy
x y
5x 
5 x
n
ex)
ex)
3
n
3
n
 125x
3
3
 2 y    2 y 
4 5
5
4 5
 32y
20
Write each expression with positive exponents only.
Negative Exponents in Numerators and Denominators
b
n
1
 n
b
ex)
ex)
and
4
2
x
3
1
n
b
n
b
1
1
 2 
4
16
1
 3
x
Use the Zero-Exponent Rule
The Zero-Exponent Property
b 1
0
ex)
ex)
1
0
3
x
0
1
Divide by using the Quotient Rule
The Quotient of Powers Property
m
b
mn

b
n
b
ex)
30
x
10
x
x
3010
x
20
Simplify by using the Quotient of Powers Rule
The Power of Quotient Property
n
a
a
   n
b
b
ex)
x
 
2
3
3
x
 3
2
n
3
x

8
Simplify.
ex)
 2x 
 
 3 
2
12
ex)
45 x
3
15 x
5
ex)
12 x y
2
 4x y
2
2 x

9
 3x
2
9
 3x y
3
0
 3x
3
Simplify.
4x  6x
2
3
ex)
12 x  4 x
2
4x
5
ex)
2
3
4x 6x


2
2
3
5
2
 2 x  3x
3
2
3
12 x 4 x

 2
2
4x
4x
 3x  x
3
SCIENTIFIC NOTATION write the answer in scientific notation.
Ex 1.)
3
7
(1.2 10 )(6.7 10 )
3
(1.2)(6.7)
(8.04)
Ex 2.)
7
(10 )(10 )
X
(10
37
3
8.04  10
=
3
(1.1)
(1.110 )

8
(5.5 10 ) (5.5)
( 0 .2 )
)
10
(10 )
8
(10 )
X
(10
3( 8)
=
)
0.2  10
5
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
5.2 Evaluate and Graph Polynomial Functions
What you should learn:
Goal 1 Evaluate a polynomial function
by using 2 kinds of Substitution
a)Direct Substitution
b)Synthetic Substitution
Goal 2
End Behavior of a function’s graph.
L1.2.1
5.2 Evaluating and Graphing Polynomial Functions
Polynomial- is a single term
or sum of two or more terms containing
variables in the numerator with whole
number exponents.
6
x6
4x
or
or
or
5x
3
4x  x  2
2
Polynomial- is a single term or sum of
two or more terms containing variables in
the numerator with whole number
exponents.
7 x  4 x  5x  6
3
2
It is customary to write the terms in the order
of descending powers of the variables.
This is Standard Form of a polynomial.
Monomials-polynomials with one term.
Example) 6 or 2x
or
4x
3
Binomials-polynomials with two terms
Example)
3x  5
Trinomials-polynomials with three terms.
Example)
4 x  5x  6
2
The Degree of
ax
n
If a does not equal zero,
then the degree of ax
n
is n.
The degree of a nonzero constant is 0.
The constant “ 0 “ has no defined degree.
Polynomial
Degree of the number is the
exponent of the variable..
Example)
Example)
ax
2x , has a degree of 1
4x
2
, has a degree of 2
Degree of the polynomial is the largest
degree of its terms.
Example)
4 x  5x  7 x  8
3
, has a degree of 3
2
n
Classifying polynomials by degree
Degree
5
Degree 0,
3x  5
Degree 1,
Leading Coef
Type
5,
Constant
4 x  5x  6
2
3,
Linear
Degree 2,
4,
Quadratic
7x
3
x  5x  7 x  6
4
2
Degree 3,
7,
Cubic
Degree 4,
1,
Quartic
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)
f ( x)  x 4  14 x 2  3
no
yes
c)
f ( x)  5x 2  3x 1  x
b)
f ( x)  7 x  3  x
2
d)
f ( x)  x  2 x  0.6 x5
no
yes
Goal 1 Evaluate a polynomial function
Directions: Use Direct Substitution to evaluate the
Polynomial Function for the given value of x.
f (x) = 2 x  8 x  5 x  7 , when x = 3
4
2
Make the Substitution.
f (3) = 2(3)  8(3)  5(3)  7
4
2
 162  72  15  7
 98
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.
2 x 4  0 x3  (8x 2 )  5x  (7)
NOTICE
Synthetic Substitution
2 x 4  0 x3  (8x 2 )  5x  (7)
Synthetic Substitution
Polynomial in standard form
2 x  0 x  (8x )  5x  (7)
4
x-value
3
3 2
0
6
2
6
add
2
-8
18
5
30
-7
105
10
35
98
Ch. 5.2 cont’ Goal 2 END BEHAVIOR OF A FUNCTION’S GRAPH
Degree: odd
Degree: odd
Leading Coefficient: positive
Leading Coefficient: negative
f (x)  
f (x )  
x


as
as x  
f (x )  
as x  
f (x)  
as x  
5.2 Evaluating and Graphing Polynomial Functions
END BEHAVIOR OF A FUNCTION’S GRAPH
Degree: even
Degree: even
Leading Coefficient: positive
Leading Coefficient: negative
f (x )  
f (x)  
as x  
as x  
f (x )  
f (x)  
as x  
as x  
Directions:
DECRIBE the degree and leading coefficient of the polynomial
function whose graph is:
a)
Degree:
b)
ODD
Leading Coef: NEG
c)
Degree:
EVEN
Leading Coef:
Degree: EVEN
POS
Leading Coef: NEG
DECRIBE THE END BEHAVIOR of the graph of the polynomial
function by completing these statements:
Ex 1)
f ( x)  ______ as
x  
f ( x)  ______ as
x  
f ( x)  13 x 5  x 3  x
Reflection on the Section
Which term of a polynomial function is most
important in determining the end behavior of the
function?
assignment
5.3 Add, Subtract , and Multiply Polynomials
What you should learn:
Goal 1
Add, subtract, and multiply
polynomials
It’s just like
Combining Like
Terms.
A1.1.4
5.3 Adding, Subtracting, and Multiplying
Add or subtract as indicated
(2 x y  xy)  (4 x y  7 xy)
2
ex)
2
2 x y  8xy
2
ex)
( x  7 xy  5 y )  (6 x  3xy  4 y )
4
3
4
 5x  4 xy  9 y
4
3
3
5.3 Adding, Subtracting, and Multiplying
Multiplying Monomials
ex)
2 x (4 x
2
multiply the coefficients
and multiply the variables
)
(2  4)( x  x )
2
8x
ex)
(3x )(4 x )  12x 9
ex)
 4 7  3 2 
12 9
 x  x  
x
35
 5  7 
6
3
3
ex)
(6 x )(2 x)( x )
4
6
 12x
11
5.3 Adding, Subtracting, and Multiplying
Finding the product of the monomial and the polynomial
2 x (4 x  3)
ex)
ex)
ex)
 8x  6x
3 y (4 y  2 y )
2
2
2
 12 y  6 y
4
3
2 y (3 y  5 y  3)  6 y  10 y  6 y
2
2
4
3
2
5.3 Adding, Subtracting, and Multiplying
Finding the product when neither is a monomial
ex)
( x  3)( 4 x  3)  4 x  3x  12 x  9
2
 4 x  15 x  9
2
ex)
( y  2)( 4 y  2 y  3)
2
 4 y  2 y  3y  8y  4 y  6
3
2
2
 4y  6y  y  6
3
2
5.3 Adding, Subtracting, and Multiplying
Find the Product
ex)
( x  y)
3
( x  y )( x  y )( x  y )
( x  2 xy  y )( x  y)
2
2
x  x y  2 x y  2 xy  xy  y
3
2
2
2
x  3x y  3xy  y
3
2
2
2
3
3
5.3 Adding, Subtracting, and Multiplying
Reflection on the Section
How do you add or subtract two polynomials?
assignment
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
Factoring Monomials means finding two monomials
whose product gives the original monomial.
x
4
x
6
2 2
(x )
3 2
(x )
2 3
(x )
Or maybe
Factoring Monomials means finding two
monomials whose product gives the
original monomial.
ex)
30x
2
Can be factored in a few
different ways…
2
a.) (5 x )( 6 x )
c.)
b.) (15 x)( 2 x)
d.) (6 x)( 5 x)
(10 x )(3)
Directions:
Find three factorizations for each monomial.
1.)
20x
3
4
2
( x )(20 x )
(4 x)(5x )
2
2
(2 x )(10 x )
2.)  15x 6
3.)
27 x
5
2
Find the greatest common factor.
1.)
6x
3
and 10x
2
2x
2
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.) 15x 5
and
27 x
3
3x
3
Factoring out the greatest common factor.
But, before we do that…do you remember the
Distributive Property?
5 x(2 x  3)
10 x  15 x
2
When factoring out the GCF, what we are going
to do is UN-Distribute.
Factor each polynomial using the GCF.
x  5x
ex)
x( x  5)
4
ex)
3
7 x ( x  3)
7 x  21x
ex)
2
15 x  5 x  10 x
3
2
5 x(3x  x  2)
2
calculator
Factoring out the GCF and then factoring
the Difference of two Squares.
Example 1)
3x  3x
3
What’s the GCF?
3x( x  1)
2
3 x( x  1)( x  1)
5.2 Solving Quadratic Equations by Factoring
Factoring out the GCF and then factoring the
Difference of two Squares.
Example 2)
12 x  3 x
3
What’s the GCF?
3x(4 x  1)
2
3x(2 x  1)( 2 x  1)
5.2 Solving Quadratic Equations by Factoring
Factor by Grouping
Ex 1)
x  2x  4x  8
3
2
( x  2 x )  (4 x  8)
3
2
x ( x  2)  4( x  2)
2
( x  2)( x  4)
2
Group into binomials
Factor-out GCF from
each binomial
Factor-out GCF
Factored by Grouping
Factor by Grouping
Ex 2)
x  2 x  9 x  18
3
2
Group into binomials
( x  2 x )  (9 x  18)
3
2
x ( x  2)  9( x  2)
2
( x  9)( x  2)
2
( x  3)( x  3)( x  2)
Factor-out GCF from
each binomial
Factor-out GCF
Factored by Grouping
Sum
Example 1)
x 8
8  23
3
A3  B3  ( A  B)( A2  AB  B 2 )
( x  2)( x  2 x  2 )
2
2
or
( x  2)( x  2 x  4)
2
Factoring Perfect Square Trinomials
Example :
x  6x  9
4
2
( x )  6( x )  9
2 2
2
x  6x  9
2
( x  3)( x  3)
( x  3)( x  3)
2
2
Since both binomials are the same you can say
( x  3)
2
2
Reflection on the Section
How can you use the zero product property to solve
polynomial equations of degree 3 or more?
assignment
6.1 nth Roots and Rational Exponents
What you should learn:
Goal 1 Evaluate nth roots of real numbers
using both radical notation and
rational exponent notation
Goal 2 Evaluate the expression.
Goal 3 Solving Equations.
6.1 nth Roots and Rational Exponents
Goal 1
Using Rational Exponent Notation
Rewrite the expression using RATIONAL EXPONENT notation.
 If n is odd, then a has one real nth root:
Ex)
3
125
a a
n
1
n
125 3
1
If n is even and a > 0, then a has two real nth roots:
Ex)
4
81
 81
1
4
 a  a
 If n is even and a = 0, then a has one nth root:
n
n
1
n
0 0 n 0
1
 If n is even and a < 0, then a has NO Real roots:
6.1 nth Roots and Rational Exponents
Using Rational Exponent Notation
Rewrite the expression using RADICAL notation.
Ex)
Ex)
24 3
3
24
28
4
28
1
1
4
6.1 nth Roots and Rational Exponents
Goal 2
Evaluating Expressions
Evaluate the expression.
Ex)
Ex)
9
3
32
2
2
 9
3
1
5
32
2

5
 3
 27
1
1
 2
2
3
 32 
5
2
1

4
6.1 nth Roots and Rational Exponents
Goal 3
Ex)
4
Solving Equations
4
x 4  81
x   81
4
x  3
When the exponent is EVEN you
must use the Plus/Minus
2 x  64
5
Ex)
5
5
x 5  32
x  32
5
x2
When the exponent is ODD you
don’t use the Plus/Minus
6.1 nth Roots and Rational Exponents
Solving Equations
Ex)
4
4
( x  4)  256
4
Take the Root 1st.
x  4   256
4
x  4  4
x4  4
x 8
Very Important
2 answers !
x  4  4
x0
6.1 nth Roots and Rational Exponents
Factoring
Given expression
GCF
Factoring
Bi-nomial
Diff of
Squares
Diff of
Cubes
Factoring
Tri-nomial
Into 2
Binomials
(
)(
)
Sum of
Cubes
Calculator
Finding the Zeros
Flaming Banana
4 term Polynomial
Grouping
Finding the REAL-Number solutions of the equation.
Quadratic
Formula
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.
a) using Long Division
b) Synthetic Division
Goal 2
Factoring using the “Synthetic Method”
Goal 3
Finding the other ZERO’s when given one of them.
A1.1.5
5.5 The Remainder and Factor Theorem
Divide using the long division
ex)
x 2  10 x  23
x3
2
x +7 
( x  3)
x  3 x  10 x  23
2
- ( x  3x )
7 x  23
- ( 7 x  21 )
2
2
6.5 The Remainder and Factor Theorem
Divide using the long division with Missing Terms
ex)
8x3  5
2x 1
4x  2x  1
3
2
2 x  1 8x  0 x  0 x  5
- (8 x 3  4 x 2 )
2
4x  0x
- (4 x 2  2 x )
2
4

(2 x  1)
2x  5
- ( 2x 1 )
4
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.
( x3  4 x 2  5x  5)  ( x  3)
3 1
4
-5
5
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
1
4
3
-5
5
5. Add the values in the
new column, writing
the sum in the bottom
row.
6. Repeat this series
of multiplications and
additions until all
columns are filled in.
3
1
4
add 3
1 7
3 1
1
-5
4
3
-5
21
7
16
5
5
add
7. Use the numbers in the
last row to write the
quotient and remainder in
fractional form.
3
1
4
add 3
1 7
The degree of the first
term of the quotient is one
less than the degree of the
first term of the dividend.
-5 5
21 48
16 53
53
x  7 x  16 
x 3
2
The final value in this row
is the remainder.
x  3 x  4 x  5x  5
3
2
Synthetic Division
To divide a polynomial by x - c
Example 1)
( x  4 x  2)  ( x  1)
2
-1
1
1
4 -2
-1 -3
3
-5
5
x 3
x 1
Synthetic Division
To divide a polynomial by x - c
Example 2)
( x  5x  7)  ( x  2)
3
2
1
1
0 -5 7
2 4 -2
2 -1 5
5
x  2x 1
x2
2
Factoring a Polynomial
Example 1)
(x + 3)
f ( x)  2 x  11x  18x  9
3
2
given that f(-3) = 0.
-3
2
11
-6
18
-15
9
-9
2
5
3
0
Because f(-3) = 0, you know that (x -(-3)) or (x + 3) is a factor of f(x).
2 x  11x  18 x  9
3
2
 ( x  3)( 2 x  5 x  3)
2
Factoring a Polynomial
Example 2)
(x - 2)
f ( x)  x  2 x  9 x  18
3
2
given that f(2) = 0.
2
1
-2
2
-9
0
18
-18
1
0
-9
0
Because f(2) = 0, you know that (x -(2)) or (x - 2) is a factor of f(x).
x  2 x  9 x  18
3
2
 ( x  2)( x  9)
2
 ( x  2)( x  3)( x  3)
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
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
The Rational Zero Theorem
factor constant t erm a 0
p

q factor of leading coefficien t a 0
Find the rational zeros of
f ( x)  x  2 x  11x  12
3
2
solution List the possible rational zeros. The leading coefficient is 1
and the constant term is -12. So, the possible rational zeros
are:
1 2 3 4 6 12
x   , , , , ,
1 1 1 1
1
1
5.6 Finding Rational Zeros
Example 1)
Find the Rational Zeros of
f ( x)  2 x  7 x  7 x  30
3
2
solution
List the possible rational zeros. The leading coefficient is 2 and
the constant term is 30. So, the possible rational zeros are:
1
3
5 15
x   , , ,
,1,2,3,5,6,10,15,30
2
2
2
2
Notice that we don’t write the same numbers twice
5.6 Finding Rational Zeros
Use Synthetic Division to decide which of the following are zeros
of the function 1, -1, 2, -2
Example 2)
f ( x)  x  7 x  4 x  28
3
-2
2
1
7 -4 -28
-2 -10 28
1 5 -14 0
f ( x)  ( x  2)( x  5x  14)
2
f ( x)  ( x  2)( x  2)( x  7)
x = -2, 2
5.6 Finding Rational Zeros
Find all the REAL Zeros of the function.
Example 3)
f ( x)  x  4 x  x  6
3
1
1
2
4
1
5
1
1
5
-6
6
6
0
f ( x)  ( x  1)( x  5x  6)
2
f ( x)  ( x  1)( x  2)( x  3)
x = -2, -3, 1
5.6 Finding Rational Zeros
Find all the Real Zeros of the function.
Example 4)
f ( x)  x  x  x  9 x  10
4
2
1
3
1
1
2
3
-1
1
1
2
1
6
-9 -10
14 10
7
3 7
-1 -2
2 5
5
0
5
-5
0
5.6 Finding Rational Zeros
-1
1
1
3 7
-1 -2
2 5
5
-5
0
f ( x)  ( x  2)( x  1)( x  2 x  5)
2
x = 2, -1
5.6 Finding Rational Zeros
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
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 > 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
Find all the ZEROs of the polynomial function.
Example 1)
f ( x)  x  5x  9 x  45
3
-5
2
1
5 -9
-5 0
1 0 -9
-45
45
0
f ( x)  ( x  5)( x  9)
2
f ( x)  ( x  5)( x  3)( x  3)
x = -5, -3, 3
5.7 Using the Fundamental Theorem of Algebra
Decide whether the given x-value is a zero of the function.
f ( x)  x  5 x  x  5 , x = -5
3
Example 1)
-5
2
1 5
-5
1 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
Write a polynomial function of least degree that has real
coefficients, the given zeros, and a leading coefficient of 1.
Example 1)
-4, 1, 5
0  ( x  4)( x  1)( x  5)
f ( x)  ( x  4)( x  1)( x  5)
f ( x)  ( x  3x  4)( x  5)
2
f ( x)  x  2 x  19 x  20
3
2
5.7 Using the Fundamental Theorem of Algebra
QUADRATIC FORMULA
 b  b  4ac
x
2a
2
Find ALL the ZEROs of the polynomial function.
Example )
f ( x)  x  3 x  2
3
2
f ( x)  ( x  1)( x  2 x  2)
2
 (2)  (2)  4(1)( 2)
x
2(1)
2
x = 2.732
x = -.732
Find ALL the ZEROs of the polynomial function.
Example #24)
f ( x)  x  2 x  x  2
3
2
f ( x)  ( x  2)( x  1)
2
Doesn’t FCTPOLY…Now what?
Find ALL the ZEROs of the polynomial function.
Example )
f ( x)  x  2 x  x  2
3
2
f ( x)  ( x  1)( x  16 x  16 x  16)
3
2
 (2)  (2)  4(1)( 2)
x
2(1)
2
Find ALL the ZEROs of the polynomial function.
Example )
-1
f ( x)  x  4 x  4 x  10 x  13x  14
5
4
3
2
1 -4
-1
4
5
10
-9
-13
-1
-14
14
1 -5
9
1
-14
0
f ( x)  ( x  1)( x  5x  9 x  x  14)
4
3
2
Graph this one….find one of the zeros..
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