Polynomials and Factors

1
Chapter 10

• Polynomials and Factors

2
Objectives

• Find the product of any two binomials.
• Factor any trinomial
• Solve Problems that involve factoring

3
Definition

• Factor a part of a mathematical expression.
• Example
• X(2x1) is the factored form of

4
Definition

• Monomial a (constant) number, a variable, or
  the product of a constant and a variable.
• Examples -3, 5x,

5
Definition

• Polynomial the sum of one or more monomials.

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Definition

• Binomial a polynomial with two terms.
• Examples
• 3x 4,

7
Definition

• Trinomial - A polynomial with three terms.
• Example

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Definition

• Like terms terms that have the same variables
  and the same exponent.

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Combining like terms

• What is the sum of

10
What is the difference
11
Adding Polynomials

• Add the coefficients of like terms.

12
Ce

• Add the following binomials

13
Ce

• Add the following polynomials

14
Subtracting Polynomials

• Must distribute the (negative sign)

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Ce

• Subtract

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Ce

• Simplify

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Page 10-8

• 5-23 odd

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Section 10-2

• Factors and Primes

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Definition

• Factor when two or more numbers and or
  variables are multiplied together. The numbers or
  variables are factors.

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• Example
• A b c
• A and b are factors
• C is the product.

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CE

• List the factors of 12

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ce

• List the factors of 14

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Definition

• Prime Numbers - a number divisible by itself and
  1.

24

• Examples of prime numbers
• 1, 2, 3, 5, 7, 11

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CE

• What are the factor pairs of 18

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Definition

• Prime factorization writing a number as a
  product of primes.

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ce

• Use a factor tree to write the prime
  factorization of 36

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Greatest common factor

• The greatest common factor of two or more
  integers is the largest factor that all the
  integers share.

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GCF

• GCF of a number is found by writing the number as
  a product of primes, and circling the common
  prime factors.

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ce

• Find the GCF of 18 and 24

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CE

• Find the GCF of 24 and 36
• Factor each into a product of primes

32
CE

• Find the GCF of 18 and 45

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CE

• Find the GCF of
• 7 and 28

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Page 10-15

• 7 - 25

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Lesson 10 - 3

• Monomial Factors

36
CEThe Distributive Property

• Distribute
• 3(24)

37
CEDistribute

• 4(52)

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Multiplying Monomials

• When multiplying monomials you
• Multiply the coefficients and add the exponents
  of like variables.

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CE

• Multiply

40
CE

• Multiply

41
Ce

• Distribute

42
Dividing Monomials

• When dividing monomials
• You subtract the exponents of like variables.

43
CeDivide
44
Ce Divide
45
CeDivide
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Page 10-19

• 9-11

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Lesson 10.4

• Zero and Negative Exponents

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Zero Property of Exponents

• For all numbers x (except 0)

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Property of negative exponents

• For all numbers x (except 0) and integers n,

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CE

• Evaluate
• In terms of a positive exponent

51
CE

• Evaluate
• In terms of a positive exponent

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Power of a product

• For all numbers x and y, and integers n,

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CE

• Evaluate

54
CE

• Evaluate

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Power of a Power

• For all numbers x and integers n and m,

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CE Simplify
57
CE Simplify
58
CE Simplify
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Page

• 10-26
• 5 - 13

60
Do worksheet
61
Section 10.5Adding Rational Expressions
62
Theorem

• If are rational
• expressions, then

63
CE

• Add

64

• When the denominators are different, you must
  find a common denominator.

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CE

• Add

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CE

• Add

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CE Simplify
68
CE Simplify
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CE

• Solve

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Page

• 10-30
• 5 13 odd

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Section 10.6

• Multiplying Binomials

72

• FOIL
• FOIL
• FOIL
• FOIL

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Multiply

• F - First
• O - Outer
• I - Inner
• L - last

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Note

• The foil method will always produce four terms,
  but the simplified the result will be a binomial
  or trinomial.

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CE

• Foil
• (x3)(x1)

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CE

• Foil
• (x1)(x2)

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CE

• Foil
• (x-2)(x-3)

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CE

• Foil
• (x-2)(x1)

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ce

• Foil
• (x-4)(x4)

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ce

• Foil
• (x5)(-x1)

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ce

• Foil
• (-x2)(-x-2)

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Page 10 38

• 5 16 odd
• Show your work

83

• Foil worksheet

84

• Foil Quiz

85
Section 10.7

• Factoring Trinomials

86
Factoring Trinomials

• This is the opposite of foil.
• We are now going to work backwards.

87
Factoring
Where c gt 0
88
To factor a trinomial you mustlist factors of c

• CE
• List the factor pairs of 12
• List the factor pairs of 18

89
Ce Factor
90
CE

• Factor

91
CE

• Factor

92
CE

• Factor

93
hw

• Page 10-41 3-11 odd

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Page 10 - 43

• Activity Factoring

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Factoring
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ce

• Factor

97
CE

• Factor

98
CE

• Factor

99
Page 10-45

• 5 16 even

100
Do factor worksheet
101
Section 10.9

• Special Products and Factors

102
The Difference of two squares

• This is a shortcut for factoring

103
Difference of Two squares

• Since
• Then

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CE

• List all of the squares from 1- 10

105
CE

• Factor

106
CE

• Factor

107
CE

• Factor

108
CE

• Factor

109
CE

• Factor

110
Page

• 10 52
• 6 22 even