1. Algebra I – CHAPTER 10 10.1 Adding and Subtracting Polynomials
Objectives:
Know the anatomy of a polynomial
Add and subtract polynomials

2. Vocabulary – PAGE 576
Polynomial Standard form Degree Degree of polynomial Leading Coefficient Term Monomial Binomial Trinomial

3. Term vs. Coefficient
Term Coefficient -12x³ ? x³y ? -z ? 2 ? Constant – If a term contains only a number like 2 it is called a constant.

4. Types of Polynomials Monomials 1 Term Binomials 2 Terms Trinomials
ax²
x + y
x² + 4xy + y²
-3x
6y² - 2
-x³ + 2x + 1 4
½ z³ - 2z
8y² - 2y -10
All other longer polynomials do not have a special name.

5. There are two ways to classify polynomials:
degree – (for each term) It is the exponent of the variable
number of terms – count the number of terms
SEE EXAMPLE 2 on Page 576

6. Polynomial Classifications
Degree
Classification
# of Terms 6
-2x + 1
x² + 2x - 5
x³ - 8x
PLEASE NOTE: X⁴ DEGREE IS QUARTIC

7. Polynomial in standard form
2x³ + 5x² - 4x + 7
Degree of a polynomial – largest degree of any of its terms.
Leading
Coefficient
Degree
Constant Term

8. Anatomy of a Polynomial
terms
Total degree
constant
degree of polynomial
standard form
leading coefficient

9. Anatomy of a Polynomial
terms
degree
constant
degree of polynomial
standard form
leading coefficient

10. Like Terms: terms that have identical variables both in letters and degree
Combining Like Terms: adding and subtracting like terms – changes only the coefficient
Adding and Subtracting Polynomials: simply combining like terms
2 ways to add and subtract
Horizontal
Vertical

11. Adding & Subtracting Horizontally
(6x2 – x + 3) + (-2x + x2 – 7)
Steps:
Distribute all subtraction signs into polynomials.
Combine like terms

12. Adding & Subtracting Vertically
(6x2 – x + 3) – (-2x + x2 – 7)
Steps:
Distribute all subtraction signs into polynomials.
Combine like terms

13. (-8x3 + x – 9x2 + 2) + (8x2 – 2x + 4) + (4x2 – 1 – 3x3)
Adding & Subtracting
(-8x3 + x – 9x2 + 2) + (8x2 – 2x + 4) + (4x2 – 1 – 3x3)
Steps:
Distribute all subtraction signs into polynomials.
Combine like terms

14. Adding & Subtracting
(-6x3 + 5x – 3) – (2x3 + 4x2 – 3x + 1)
Steps:
Put each polynomial on a level in standard form lining up like terms
Carefully add or subtract the coefficients

15. Adding & Subtracting
(4x2 – 1) – (3x – 2x2)
Steps:
Put each polynomial on a level in standard form lining up like terms
Carefully add or subtract the coefficients

16. (-8x3 + x – 9x2 + 2) + (8x2 – 2x + 4) – (4x2 – 1 – 3x3)
Adding & Subtracting
(-8x3 + x – 9x2 + 2) + (8x2 – 2x + 4) – (4x2 – 1 – 3x3)
Steps:
Put each polynomial on a level in standard form lining up like terms
Carefully add or subtract the coefficients

17. (2x2 + 9x – 4) + (6x – 3x2 + 1) Find the sum or difference: Steps:
Put each polynomial on a level in standard form lining up like terms
Carefully add or subtract the coefficients

18. Algebra I – Lesson 10.1
Homework P. 580 #’s 19-23, 47-52

19. Algebra I – Chapter 10 10.2 Multiplying Polynomials Objectives:
Objectives:
Multiply 2 binomials
Multiply polynomials
Vocabulary:
none

20. Multiplying Polynomials
Multiplying variable terms: Multiply the coefficients together and then multiply the variables together. Then combine like terms, if necessary.
3x(6xy) =
18x²y
2x²(5y² + 7xy) =
10x²y² + 14x³y

21. Adding vs. Multiplying Polynomials
Variable exponents must be the same to perform operation.
Coefficients must be the same to perform operation.
Variable and exponents will stay the same after operation.
Variable
exponents
change after operation
Adding
Polynomials
Yes No
Multiplying
No*
*Unless:
7(3x²) = 21 x²
Yes*

22. Multiplying Binomials using FOIL
(3x + 4)(x + 5) = 3x² + 15x + 4x + 20
= 3x² + 19x + 20
First Terms
Inner Terms
Last Terms
Outer Terms

23. Multiplying Polynomials Horizontally
Find the product (x – 4)(5x x²)

24. Multiplying Polynomials Vertically
Find the product (x + 4)(5x x²)
Align like terms (descending order) in columns.
- 3x² + 5x + 3
* x + 4

25. (x + 9)(x + 7) (x + 5)(x + 5) (x + 7)(x – 7) (2x + 3)(2x +3)+ 3)
Multiply these polynomials:
(x + 9)(x + 7) (x + 5)(x + 5)
(x + 7)(x – 7) (2x + 3)(2x +3)+ 3)

26. ((2x – 3)(3x + 2) (3x2 + 3)(5x2 – 2x) (5x - 3)(5x2 – 3x + 2)
Multiply these polynomials:
((2x – 3)(3x + 2) (3x2 + 3)(5x2 – 2x)
(5x - 3)(5x2 – 3x + 2)

27. Algebra I – Lesson 10.2
Tonight’s Homework P. 587 #’s 23-32

28. Algebra I – Chapter 10 10.3 Special Products of Polynomials
Objectives:
Multiply using the square of a binomials pattern
Multiply using the sum and difference pattern

29. Find the pattern: Use FOIL to find the products: (2n +3)(2n - 3)
(x - 2)(x+2)
(2n +3)(2n - 3)
(4t – 1)(4t + 1)
(x + y)(x – y)
x² - 4
4n² - 9
16t² - 1
x² - y²

30. Sum and Difference Pattern
(a + b)(a – b) = a² - b² (3x – 4) (3x + 4) = 9x² - 16
Ex. 1 (2x -5)(2x + 5) =
Ex. 2 (x - 2y)(x + 2y) =

31. Multiply:
(x + 3)(x - 3)

32. Multiply:
(x - 4) (x + 4)

33. Multiply:
(x - 5) (x + 5)

34. Multiply:
(2x + 7) (2x - 7)

35. Multiply:
(3x - 4) (3x + 4)

36. Multiply:
(3x3 + 11)(3x3 - 11)

37. Find the pattern: Use FOIL to find the products: (x + 3)² (3m + 1)²
(x + y)²
x² + 6x + 9
9m² + 6m + 1
25s² + 20s + 4
x² + 2xy + y²

38. Square of a Binomial Pattern
(a + b)² = a² + 2ab + b²
Ex: (x + 4)² = x² + 8x + 16
(a - b)² = a² - 2ab + b²
Ex: (2x – 6)² = 4x² - 24x + 36
9b² - 30b + 25
(3b -5)² =

39. Multiply:
(5x2 + 4)2

40. Multiply:
(2x2 – 3y)2

41. Multiply:
(5x4 – 3x2)2

42. Algebra I – Lesson 10.3 Tonight’s Homework P. 593 #’s 15-33 ODD ONLY
Quiz FRIDAY – STUDY!

43. 10.1-10.3 Review Complete Both Problems on your OWN!
(6x + 2x³ - 5x⁵) – (9x³ + 2x⁵ - 4)
(6x + 2)(4x – 3)

44. Algebra I –Chapter 10
10.4 Solving Polynomial Equations in Factored Form
Objectives:
Solve a polynomial equation in factored form
Understand factors, solutions, and zeros
Vocabulary:
factored form, zero-product form

45. Vocabulary
We use evaluate, simplify, and find the product for expressions.
In order to solve there must be an equation. Equations have the = sign.
A polynomial in factored form is written as a product of two or more linear factors. The factored form of x² +4x + 4 is (x + 2)(x + 2).

46. Standard form:
2x2 + 7x – 15 = 0
Factored form:
(2x – 3)(x + 5) = 0

47. Zero-Product Property
Let a and b be real numbers. If ab = 0, then a = 0 or b = 0
Solve the equation (x – 4)(x + 1) = 0
By the zero product property:
x – 4 = 0 or x + 1 = 0
x= or x = -1
The solutions are 4 and -1. Check in original equation.

48. Solve the following equations.-8
4, -3
0, -3, 3/2
2/3, -3/4, -4
Solve (x + 8)² = 0
Solve (x – 4)(x + 3) = 0
Solve x(x+3)(2x – 3) = 0
(3x – 2)(4x + 3)(x + 4) = 0

49. Solve the following equations.
(x – 9)(x + 6) = 0
(5x – 2)(2x + 7) = 0
(x + 8)2 = 0
(3x – 2)(4x + 3)(x + 4) = 0
(x – 3)(5x + 19)(5x – 1) = 0

50. Factors, Solutions, and x intercepts
For any quadratic polynomial ax² + bx + c, if one statement is true, then all 3 statements are true.
(x – 4) and (x + 4) are factors of x² -16
x = 4 and x = -4 are solutions of x² - 16 = 0
4 and -4 are x-intercepts of y = x² - 16

51. Sketch the graph of y = (x – 3)(x + 1)
Solve (x – 3) (x + 1) = 0 x intercepts = 3, -1 Coordinates of the vertex: The x coordinate is the avg. of the x intercepts x = (3 + (-1)) / 2 = 1 Substitute to find y- coordinate y = (1 – 3)(1 + 1) y = (-2)(2) = -4 The coordinates of the vertex are (1, -4).
(3,0)
(-1,0)
(1,-4)

52. Sketch the graph of y = (x – 4)(x + 2)
Solve (x – 4) (x + 2) = 0 x intercepts = Coordinates of the vertex: The x coordinate is the avg. of the x intercepts Substitute to find y- coordinate The coordinates of the vertex are ____

53. Algebra I – Lesson 10.4
Tonight’s Homework P #’s 19-23, 34-35, 44-47

54. Algebra I – chapter 10 10.5 Factoring x2 + bx + c none Objectives:
Objectives:
Factor x2 + bx + c
Solve x2 + bx + c = 0 equations by factoring
Vocabulary:
none

55. (x + one number) (x + other number)
Factoring x² + bx + c
Find 2 numbers whose product is c and whose sum is b. The factored form is:
(x + one number) (x + other number)

56. Factoring when b and c are positive
c b
2 3
1·2 1+2
x² + 3x + 2 (x + ) (x + ) Fill in numbers from table. (x + 1)(x + 2) Check by using FOIL: (x + 1)(x + 2) = x² + 1x + 2x + 2 = x² + 3x + 2

57. (x + 2) (x + 8) Factor: x² + 10x + 16 c b +
1, 16
2, 8
4, 4 17 10 8
(x + 2) (x + 8)
Check by using foil: (x + 2) (x + 8) = x² + 8x + 2x + 16 = x² + 10x + 16

58. Factor x² - 5x + 6 6 -5 + (x + -2) (x + -3) = (x - 2) (x - 3)
Since c is a positive number we will look for 2 numbers that have the same signs. The 2 numbers will be - since b is -. +
-1, -6
-2, -3
-3, -2
-6, -1 -7 -5
(x + -2) (x + -3) = (x - 2) (x - 3)
Check by foil:
(x – 2) (x - 3) = x² - 2x - 3x + 6 = x² - 5x + 6

59. Factor x² - 2x - 8 -8 -2 + (x + -4) (x +2) = (x – 4) (x + 2)
Since c is a negative number we will look for 2 numbers that have opposite signs. The larger number will be - since b is -. -2 +
-1, 8
-2, 4
-4, 2
-8, 1 7 2 -2 -7
(x + -4) (x +2) = (x – 4) (x + 2)
Check by foil: (x – 4) (x + 2) = x² - 4x + 2x – 8 = x² - 2x - 8

60. x² - bx + c x² + bx - c x² - bx - c Sign patterns
+ means factors have the same sign
x² - bx + c
x² + bx + c
Both numbers will be -
Both numbers will be +
- means factors have different signs
x² + bx - c
x² - bx - c
Larger abs value will be -
Larger abs value will be +

61. YOU TRY to Factor: x² + 7x - 18
-18 7
Since c is a negative number we will look for 2 numbers that have opposite signs. The larger number will be + since b is +. +

62. Factor Each Expression:
x2 – 2x - 8
x2 – 11x + 30
x2 – 5x + 6
x2 – 12x - 64
x2 + 17x + 72
x2 + 10x - 39
x2 - 6x - 27

63. Solving a Quadratic Equation
x² - 3x = 10 Equation x² - 3x – 10 = 0 Write in Standard Form (x - 5)(x + 2) = 0 Factor left side x–5=0 or x+2= 0 Zero Product Property X = 5 or x = -2 Solve for x The solutions are 5 and -2. Check by substituting values into original equation.
(-2)² - 3 (-2) = 10
4 – (-6) = 10
5² - 3(5) = 10
25 – 15 = 10

64. Solve:
x2 + 3x = 10
x2 – 5x - 24 = 0
Steps to solve quadratic by factoring
Put into standard form
Factor quadratic
Set each factor to 0 and solve
Check!

65. x2 + 5x = -6 Solve: x2 + 3x = 10 Steps to solve quadratic by factoring
Put into standard form
Factor quadratic
Set each factor to 0 and solve
Check!

66. Algebra I – Lesson 10.5
Tonight’s Homework P. 607 #’s 15-21, 27-30

67. 10.4 Review
Factor and Solve on your OWN!
x² - 10x + 24 = 0

68. Algebra I – chapter 10 10.6 Factoring ax2 + bx + c none Objectives:
Objectives:
Factor ax2 + bx + c
Solve ax2 + bx + c = 0 equations by factoring
Vocabulary:
none

69. Factor: 2x² + 11x + 5 a b c
Step 1: Put in standard form and factor out GCF if possible.
Step 2: Find factors of a•c that add up to b:
Step 3: Write as factors: (ax + )(ax + ):
(2x + 1)(2x + 10)
Step 4: Factor out and discard a. Leave whole numbers.
(2x + 1)(2x + 10) = (2x + 1)(x + 5)
a•c 10 b 11
2,5 7
1,10 2 2

70. Factor: 3x² - 4x - 7 a c b
Step 1: Put in standard form and factor out GCF if possible.
Step 2: Find factors of a•c that add up to b:
Step 3: Write as factors: (ax + )(ax + ):
(3x + 3)(3x - 7)
Step 4: Factor out and discard a. Leave whole numbers.
(3x + 3)(3x - 7) = (x + 1)(3x - 7)
a•c
-21 b -4
1, -21
-20
3, -7 3 3

71. Factor: 6x² - 19x + 15 a b c
Step 1: Put in standard form and factor out GCF if possible.
Step 2: Find factors of a•c that add up to b:
Step 3: Write as factors: (ax + )(ax + ):
Step 4: Factor out and discard a. Leave whole numbers.

72. You try to Factor: 2x² + 3x - 9b c
Step 1: Put in standard form and factor out GCF if possible.
Step 2: Find factors of a•c that add up to b:
Step 3: Write as factors: (ax + )(ax + ):
Step 4: Factor out and discard a. Leave whole numbers.

73. YOU TRY - Solve for x: 6x² - 2x - 8
a•c
-12 b -1
Step 1: Put in standard form and factor out GCF if possible: 2(3x² – x – 4)
Step 2: Find factors of a•c that add up to b:
Step 3: Write as factors: (ax + )(ax + ):
Step 4: Factor out and discard

74. Factor: 15x² - 27x – 6 = 0 a b c
Step 1: Put in standard form and factor out GCF if possible.
Step 2: Find factors of a•c that add up to b:
Step 3: Write as factors: (ax + )(ax + ):
Step 4: Factor out and discard a. Leave whole numbers.

75. YOU TRY ---- Factor: 8x² - 22x + 5 = 0b c
Step 1: Put in standard form and factor out GCF if possible.
Step 2: Find factors of a•c that add up to b:
Step 3: Write as factors: (ax + )(ax + ):
Step 4: Factor out and discard a. Leave whole numbers.

76. Algebra I – Lesson 10.6 Tonight’s Homework P. 614-615 #’s 18-22, 49-51
QUIZ ON FRIDAY!

77. Algebra I – chapter 10 10.7 Factoring Special Products GCF Objectives:
Objectives:
Factor out the GCF
Factor special products
Vocabulary:
GCF

78. a² - b² = (a + b)(a – b) 9x² - 16 = (3x + 4)(3x – 4)
Difference of 2 Squares
a² - b² = (a + b)(a – b)
9x² - 16 = (3x + 4)(3x – 4)
Both binomial terms are perfect squares with a “–” between positive terms.

79. Solve for x: 2x² - 72 = 0
Put in standard form and factor out GCF: 2(x² - 36) = 0 Check for pattern and factor: 2(x + 6)(x – 6) = 0 Use zero product property: x = -6, 6 Check in original equation: 2(6)² - 72 = 0 2 (-6)² - 72 = 0
a = x
b = 6

80. Solve for x: 2x² - 72 = 0
If you forget formula, you can still factor using product-sum method by rewriting with 0x.
After GCF: 2(x² - 36) = 0
Put in standard form: 2(x² + 0x – 36) = 0
Rewrite as factored form:
2(x + 6)(x – 6) = 0
Solve using zero product property:
x = 6, -6
a•c b a c b
-36
6,-6

81. Perfect Square Trinomials
a² + 2ab + b² = (a + b)² x² + 8x + 16 = (x + 4)² a² - 2ab + b² = (a - b)² x² - 12x + 36 = (x – 6)²

82. Factoring Perfect Square Trinomials
First and last terms are perfect squares.
Middle term is 2• √first term •√last term.
x² - 4x + 4
16y² + 24y + 9
3x² - 30x + 75

83. You Try
1. Factor each Expression.
100x² x² -12x x²
+ 36x = -108
2. Use the Distributive Property to simplify.
3x(4x + 9) (-3)(2x³ + 5)

84. Algebra I – Lesson 10.7
Tonight’s Homework P. 622 #’s 18-22, 27-30, 35

85. Algebra I – chapter 10 10.8 Factoring Using the Distributive Property
Objectives:
Factor out the GCF
Solve equations with degrees > 2
Vocabulary:
none

86. Binomials: Both terms must be perfect squares separated with a – sign.
a) m² b) 4p² - 25
c) 50 – 98x² d) x² + 36

87. A polynomial is prime if it can not be factored to polynomials having integer coefficients.
Factor Completely:
75x – 3x² w + 21w³
5x³ - 25x² + 30x

88. YOUR TURN: Always Factor GCF First
33x x²
4p² + 12p
18d³ - 6d² + 3d

89. Factoring by Grouping x³ + 2x² + 3x + 6 No GCF (x³ + 2x²) + (3x + 6)
Split to group terms
Factor each group
Distributive Property

90. TOGETHER Factor by Grouping
x³ + 2x² - 36x – 72 = 0

91. YOUR TURN Factor by Grouping
x³ - 2x² - 9x + 18

92. ONE MORE Factor by Grouping
x³ - 3x² + x – 3

93. Summary of Factoring a Polynomial
Step 1: Factor out GCF
Step 2: 2 Terms: Factor a² - b²
3 Terms: Factor x² + bx + c
Factor ax² + bx + c
4 Terms: Factor by Grouping
Step 3: Check by multiplying back together.
Can it be factored further?

94. Tonight’s Homework P. 629 #’s 18-22, 27-28, 30, 33 TEST NEXT TUESDAY!
Algebra I – Lesson 10.8
Tonight’s Homework
P. 629
#’s 18-22, 27-28, 30, 33
TEST NEXT TUESDAY!