2. 1.1 – Integer Exponents
• For any real number b and any natural
number n, the nth power of b is found by
multiplying b as a factor n times.
n
b b b b b
3. Exponential Expression – an
expression that involves
exponents
• Base – the number being multiplied
• Exponent – the number of factors of the
base.
14. Def: Monomial
• An expression that is a constant
or a product of a constant and
variables that are raised to
whole –number powers.
• Ex: 4x 1.6 2xyz
15. Definitions:
• Coefficient: The numerical
factor in a monomial
• Degree of a Monomial: The
sum of the exponents of all
variables in the monomial.
23. To add or subtract Polynomials
• Combine Like Terms
• May be done with columns or
horizontally
• When subtracting- change the
sign and add
24. Evaluate Polynomial Functions
• Use functional notation to
give a polynomial a name
such as p or q and use
functional notation such as
p(x)
• Can use Calculator
25. Calculator Methods
• 1. Plug In
• 2. Use [Table]
• 3. Use program EVALUATE
• 4. Use [STO->]
• 5. Use [VARS] [Y=]
• 6. Use graph- [CAL][Value]
29. Procedure: Multiply a
polynomial by a monomial
• Use the distributive property to
multiply each term in the
polynomial by the monomial.
• Helpful to multiply the
coefficients first, then the
variables in alphabetical order.
32. Procedure: Multiplying
Polynomials
• 1. Multiply every term in the
first polynomial by every term
in the second polynomial.
• 2. Combine like terms.
• 3. Can be done horizontally or
vertically.
43. Procedure: Determine greatest common
factor GCF of 2 or more monomials
• 1. Determine GCF of numerical
coefficients.
• 2. Determine the smallest
exponent of each exponential
factor whose base is common to
the monomials. Write base with
that exponent.
• 3. Product of 1 and 2 is GCF
44. Factoring Common Factor
• 1. Find the GCF of the terms
• 2. Factor each term with the
GCF as one factor.
• 3. Apply distributive property
to factor the polynomial
46. Factoring when first terms is
negative
• Prefer the first term inside parentheses to be
positive. Factor out the negative of the
GCF.
3
2
20 36
4 (5 9)
xy y
y xy
47. Factoring when GCF is a
polynomial
( 5) ( 5)
( 5)( )
a c b c
c a b
48. Factoring by Grouping – 4 terms
• 1. Check for a common factor
• 2. Group the terms so each group has a
common factor.
• 3. Factor out the GCF in each group.
• 4. Factor out the common binomial factor –
if none , rearrange polynomial
• 5. Check
49. Example – factor by grouping
2 2
32 48 20 30
xy xy y y
2 16 24 10 15
y xy x y
2 2 3 8 5
y y x
50. Ralph Waldo Emerson – U.S.
essayist, poet, philosopher
•“We live in
succession , in
division, in parts, in
particles.”
61. Factoring
• 1. Find two numbers with a product equal
to c and a sum equal to b.
• The factored trinomial will have the form(x
+ ___ ) (x + ___ )
• Where the second terms are the numbers
found in step 1.
• Factors could be combinations of positive
or negative
2
x bx c
62. Factoring
Trial and Error
• 1. Look for a common factor
• 2. Determine a pair of coefficients of first
terms whose product is a
• 3. Determine a pair of last terms whose
product is c
• 4. Verify that the sum of factors yields b
• 5. Check with FOIL Redo
2
ax bx c
63. Factoring ac method
• 1. Determine common factor if any
• 2. Find two factors of ac whose sum is b
• 3. Write a 4-term polynomial in which by
is written as the sum of two like terms
whose coefficients are two factors
determined.
• 4. Factor by grouping.
2
ax bx c
64. Example of ac method
2
6 11 4
x x
2
6 3 8 4
x x x
3 (2 1) 4(2 1)
x x x
(2 1)(3 4)
x x
65. Example of ac method
2 2
5 (8 10 3)
y y y
2 2
5 8 2 12 3
y y y y
2
5 2 4 1 3 4 1
y y y y
2
5 4 1 2 3
y y y
66. Factoring - overview
• 1. Common Factor
• 2. 4 terms – factor by grouping
• 3. 3 terms – possible perfect square
• 4. 2 terms –difference of squares
• Sum of cubes
• Difference of cubes
• Check each term to see if completely
factored
67. Isiah Thomas:
• “I’ve always believed no
matter how many shots I
miss, I’m going to make
the next one.”
70. Example of zero factor property
5 2 0
5 0 2 0
5 2
5,2 2, 5
x x
x or x
x or x
or
71. Solving a polynomial equation by
factoring.
1. Factor the polynomial
completely.
2. Set each factor equal to 0
3. Solve each of resulting equations
4. Check solutions in original
equation.
5. Write the equation in standard
form.
72. Example – solve by factoring
2
3 11 4
x x
2
3 11 4 0
x x
3 1 4 0
x x
3 1 0 4 0
x or x
1
4
3
x or x
73. Example: solve by factoring
3 2
3 2
2
4 12
4 12 0
4 12 0
6 2 0
0,6, 2
x x x
x x x
x x x
x x x
74. Example: solve by factoring
• A right triangle has a
hypotenuse 9 ft longer than the
base and another side 1 foot
longer than the base. How long
are the sides?
• Hint: Draw a picture
• Use the Pythagorean theorem
75. Solution
• Answer: 20 ft, 21 ft, and 29 ft
2 2
2
1 9
x x x
20 4
x or x
76. Example – solve by factoring
• Answer: {-1/2,4}
3 2 7 12
x x
77. Example: solve by factoring
• Answer: {-5/2,2}
2 2
1 1 1
3 2
2 12 3
x x x
78. Example: solve by factoring
• Answer: {0,4/3}
2
9 1 4 6 1 3
y y y y y
79. Example: solve by factoring
• Answer: {-3,-2,2}
3 2
3 13 7 3 1
t t t t
80. Sugar Ray Robinson
• “I’ve always believed that
you can think positive just
as well as you can think
negative.”
81.
82. Maya Angelou - poet
• “Since time is the one
immaterial object which we
cannot influence – neither
speed up nor slow down, add
to nor diminish – it is an
imponderably valuable gift.”
