1. Chapter 5A: Polynomials
Algebra 2A -2012

2. Lesson 5.1A I can identify the base and the exponent.
I can simplify product of monomials.

3. bn Power What is it? Examples: Important characteristic: Nonexamples:
Base: ____
Exponent: ____
Shortcut for repeated multiplications

4. (45)(42) (6x2)(x4) (3x4y)(-4x2) Example 1: Simplify the following
a. b. c.

5. (23)(24) (5v4)(3v) (-4ab6)(7a2b3) Your Turn 1: Simplify the following

6. bn • bm = bn + m same bases Product of Powers x3 • x4 (2x3y)(5x)
What is it?
Examples:
x3 • x4
bn • bm = bn + m
same bases
(2x3y)(5x)
Product of Powers
Important characteristic:
Nonexamples:
Add the
exponents
x3 • x4 ≠ x12

7. (x5)2 (y7)3 I can simplify product of Powers.
Example 2: Simplify the following
a. b. c. (3xy)2
(x5)2
(y7)3

8. Your Turn 2: Simplify the following
(2b2)4
(4abc)2

9. Example 3: Simplify the following
(10ab4)3 (3b2)2 d.

10. Your Turn 3: Simplify the following
(2xy2)3 (-4x5)2 4.

11. I can simplify quotient of monomials.
Example 4
a. 16x3y4
4 x5y b.

12. Your Turn!
(3xy5)2
(2x3y7)3

13. Homework: Lesson 5.1 A
Check your answers !

14. Lesson 5.1B I can simplify quotient of monomials.
I can simplify monomials with negative exponents.
I can simplify monomials with a zero exponent.

15. (5xy)0 = 1 (5xy)0 ≠ 0 (b)0 = 1 Zero Exponents 8-2 Anything to the
power of zero is one!
(5xy)0 ≠ 0

16. (b)-n = (5a)-1 = (5a)-1 ≠ -5a Negative Exponents 8-2
“move” up or down to
make it a positive exp.
(5a)-1 ≠ -5a

17. Negative exponent rule:
a-n =
a-n is the reciprocal of an
Example1:
1. x a-2b a-7b-3

18. Your Turn 1: a. 3w-3 b. 4x-7 c. 3x0y-4

19. Example 2: 4. 5.

20. Your Turn 2: d. e. f.

21. Learning target(s):
I can multiply and divide monomials. (LT1)
Mixed practice:
Simplify expressions means to write expressions without parentheses or negative exponents. a. b. c.

22. Learning target(s):
I can multiply and divide monomials. (LT1)
Mixed practice: d. e. f.

23. Learning targets:
I can multiply and divide monomials. (LT1)
Mixed practice: g. h.

24. Learning target(s):
I can multiply and divide monomials. (LT1)
Mixed practice:
Simplify expressions means to write expressions without parentheses or negative exponents. a. b. c.

25. Learning target(s):
I can multiply and divide monomials. (LT1)
Mixed practice: d. e. f.

26. Learning targets:
I can multiply and divide monomials. (LT1)
Mixed practice: g. h.

27. Homework: Lesson 5.1B

28. Warm- up over Lesson 5.1A & B
Simplify the following. 1. 2. 3. 4. 5.
6. Write a problem involving operations with powers whose answer is: -8x3

29. 5.1B

31. Lesson 5.2 (LT3) I can recognize a monomial, binomial, or trinomial.
I can identify the degree of a polynomial.
I can rewrite polynomials in descending order.
I can add polynomials.
I can subtract polynomials.
I can multiply polynomials.

32. Definition of monomial

33. Classify as monomial or non-monomial
Example 1a:
Examples:
Non Examples:

34. Definition of Polynomials

35. Classify as polynomial or non-polynomial
Example 1b:
Examples:
Non Examples: 51

36. a. 2x3 + 4x2 4 b. 2xy3 – 4x4y0 + 2x c. x2 + 3xy y d. 4x-2
Your Turn 1: Is it a polynomial? Yes or No?
Classify as monomial, binomial, or trinomial?
a. 2x3 + 4x2 4
b. 2xy3 – 4x4y0 + 2x
c. x2 + 3xy y
d. 4x-2

37. Descending: decreasing (biggest to smallest)
Example 2: Arrange the terms of the polynomial so that the
powers of x are in descending order
Descending: decreasing (biggest to smallest)
a. 4x2 + 7x3 + 5x
b. 9x3y – 4x5 + 8y - 6xy4
Your turn 2:
c x3y – 2x4y2 + 8x

38. a. 5x2 b. -9x3y5 Your Turn 3: c. 7xy5z4 d. 10
Example 3: Find the degree of the monomial.
Degree of a monomial: sum of the exponents of the variables
a. 5x2
b. -9x3y5
Your Turn 3:
c. 7xy5z4
d. 10

39. a. 5x4 + 3x2 – 9x Your Turn 4: b. 4x3 – 7x + 5x6
Example 4: Find the degree of the polynomial.
Degree of a polynomial: it is the highest degree after finding the degree of each term.
a. 5x4 + 3x2 – 9x
Your Turn 4:
b. 4x3 – 7x + 5x6

40. Check for Understanding Lesson 5.2 A: (LT3)
Is the polynomial a monomial, binomial, or trinomial.
a. y3 – 4 b. 3y3 + y0 – 2x c. -4xy5
Arrange the terms of the polynomial so that the powers of x are in descending order.
d. 3x4y3 – x6y0 + 6 – 2x
Find the degree of the monomial
e x4y3z f ab0y3
Find the degree of the polynomial.
g. 3x4y3 – x6y0 + 6 – 2x

41. (3x2 – 4x + 8) + (2x -7x2 -5) Example 5: Simplify the following
(5y2 – 3y + 8) + (4y2 – 9)
Your Turn 5: Simplify the following (LT4)
(3x2 – 4x + 8) + (2x -7x2 -5)

42. (3x2 – 4x + 8) – (2x -7x2 -5) 3. (2x2 – 3x + 1) – (x2 + 2x – 4)
Example 6: Simplify the following
(2x2 – 3x + 1) – (x2 + 2x – 4)
4. (4y2 – 9) – (5y2 – 3y + 8)
Your Turn 6: Simplify the following (LT4)
(3x2 – 4x + 8) – (2x -7x2 -5)

43. Example 7: Simplify the following.
y( 7y + 9y2)
x(2x2 + xy – 3)

44. Your Turn 7: Simplify the following. 1. n2(3n2 + 7)
n2(3n2p - np + 7p)

45. Example 8: Multiply the following.
1. (x + 3)(x + 2)
2. (x  9)(3x + 7) 3.

46. Your Turn 8: Find the product. (LT5)a. b.

47. Simplify. c. d. e.

48. Homework:
Practice 5.2

51. Mixed Review Lessons 5.1 & 5.2:
Simplify the following
(-2a2 b0)3 =
x2y(-4x + 2y3) =
(2x2 – 3x – 10) – (3x + 4x2 – 4)
Find the degree of the polynomial.
x2yz4 – 7x4y3z

52. Mixed Review Lessons 5.1 & 5.2
Simplify the following.
(3x + 2y) + (3x – 2y)
(3x + 2y) – (3x – 2y)
x(3x + 2y)
(3x + 2y)(3x – 2y)

53. Mixed Review Lessons 5.1 & 5.2
Simplify the following
(x3 y 2)(x 4 y6) =
(32x5 y8) 2 =
11. (4x 2 – 8x + 10) – (x 2 – 12)
12.
13.
Find the degree of the polynomial.
14. 3x2y5 – 10xy8

54. Simplify the following. 15.
16.
17.
18.
19.
20. Write a problem involving operations with polynomials whose answer is:
9x2 – 30x + 25

55. Dividing Polynomials lesson 5-3 A
Learning targets:
I can divide polynomials by a monomial.(LT6)
I can divide polynomials using long division. (LT7)
Notes first!
We will start the Quiz at

56. Expressions that represent division: Non-examples:

57. To divide a polynomial by a monomial,
use the properties of powers from lesson 5-1.
Example 1: Simplify.

58. Example 2: Simplify

59. Your Turn 1: Simplify

60. Review: How to divide using long division.

61. To divide by a polynomial by a polynomial,
use a long division pattern.
Remember that only like terms can be added or subtracted.
Example 3: Use long division to find the following.

62. Example 4: Use long division to divide.

63. Example 5:

64. Your Turn 2: Divide using long division.

65. Your Turn 2: Divide using long division.

66. Your Turn 2: Divide using long division.

68. Closure Lesson 5.3a: Dividing Polynomials
Homework: Practice 5.3 part I

69. Closure Lesson 5.3a: Dividing Polynomials
Homework: Practice 5.3 part I

70. Simplify the following. 1.
Warm- up Lesson 5.3a
Simplify the following. 1. 2. 3. 4.
5. Write a problem involving division of polynomials whose answer is:
x - 3

71. Part I

73. Dividing Polynomials lesson 5-3 B
Learning targets:
I can divide polynomials using synthetic division. (LT8)

74. Use Synthetic Division: A procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x – r.
Example 6: Use synthetic division to divide.

75. Use Synthetic Division: A procedure to divide a polynomial
by a binomial using coefficients of the dividend and the
value of r in the divisor x – r.
Example 7: Use synthetic division to divide.

76. Your Turn 3: Divide using synthetic division.

77. Closure Lesson 5-3: Dividing Polynomials
Homework: Practice 5.3 part II (worksheet)

78. Closure Lesson 5-3: Dividing Polynomials
Homework: Practice 5.3 part II (worksheet)

79. (4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division
Warm- up Lesson 5.3 part II
(4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division
(x3 - 4x2 + 6x – 4) ÷ (x – 2) synthetic

80. Part II

83. (4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division
Warm- up Lesson 5.3 part II
Alg. 2B
(4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division
2. (x3 - 4x2 + 6x – 4) ÷ (x – 2) synthetic
2y – y y3 – 5y y + 4

84. Part II

86. Factoring Polynomials lesson 5-4 A
Learning Targets:
I can factor polynomials using GCF.
I can factor polynomials with four terms by grpouping.
I can factor trinomials.
I can factor polynomials with two terms.

88. Technique
It means…
Example…
GCF
Grouping
Trinomials
PST
Difference of squares
Sum of Cubes
Difference of Cubes

89. Factoring by GCF
I can factor a polynomial by using Greatest Common Factor.

90. 3(a + b) = 3a + 3b = x(y – z) = xy – xz = 6y(2x + 1) = 12xy + 6y =
Simplifying
Factoring
3(a + b) =
3a + 3b =
x(y – z) =
xy – xz =
6y(2x + 1) =
12xy + 6y =

91. Example 1: Factor by using Greatest Common Factor (GCF).
Find the GCF of each term.
Use the GCF to find the remaining factors.

92. Your Turn 1: Factor by using Greatest Common Factor (GCF).
Find the GCF of each term.
Use the GCF to find the remaining factors.

93. Example 2:
Factor by using Greatest Common Factor (GCF).
4x2y – 10x3y4
Your Turn 2:
Factor by using Greatest Common Factor (GCF).
14ab3 – 8b3

94. 2x4y2 – 16xy + 24x3y3 15a3b2 + 10a2b4 – 25ab3 Example 3:
Factor by using Greatest Common Factor (GCF).
2x4y2 – 16xy + 24x3y3
Your Turn 3:
Factor by using Greatest Common Factor (GCF).
15a3b2 + 10a2b4 – 25ab3

95. Factoring by grouping
I can factor polynomials with FOUR terms by grouping .

96. Example 4: Factor by grouping.

97. Example 5: Factor by grouping.

98. Your Turn 4: Factor the following by grouping or using the area model.

99. Warm up:
Simplify:
(x3 – 2x2 + 2x – 6) ÷ (x - 3) use synthetic division
(3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) use long division
Factor:
7xy3 – 14x2y5 + 28x3y2 using GCF
ab – 5a + 3b -15 using grouping

100. Warm up/Review Simplify:
(x3 – 2x2 + 2x – 6) ÷ (x - 3) use synthetic division
(3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) use long division
Factor:
7xy3 – 14x2y5 + 28x3y2 using GCF
ab – 5a + 3b -15 using grouping
Review: Simplify
5) (x – 3y)(x + 3y)
6) (y – 2)(y + 5)
7) (5 + 2x) + (-1 – x)
8) (-7 – 3x) – (4 – 3x)

101. Warm up Day 6-Answers Simplify:
(x3 – 2x2 + 2x – 6) ÷ (x - 3) x2 + x /(x-3)
(3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) x3 + x2 - 2x /(3x-2)
Factor:
7xy3 – 14x2y5 + 28x3y2 7xy2(y – 2xy3 + 4x2)
ab – 5a + 3b -15 (a+3)(b – 5)
Review: Simplify
5) (x – 3y)(x + 3y) x2 – 9y2
6) (y – 2)(y + 5) y2 + 3y - 10
7) (5 + 2x) + (-1 – x) x
8) (-7 – 3x) – (4 – 3x) -11

102. Part I

103. Factoring Trinomials lesson 5-4 A
I can factor a trinomial.

105. Technique
It means…
Example…
GCF
Grouping
Trinomials
PST
Difference of squares
Sum of Cubes
Difference of Cubes

106. whose __________ is __________ .
Find factors (product) of ____________
whose __________ is __________ .

107. Example 1: Factor the trinomial.
a b.

108. Your Turn 1: Factor the trinomial.
a b.

109. Find factors (product) of a & c whose sum is b .
REPLACE bx using those two numbers and factor by grouping.

110. Example 2: Factor the trinomials.a.

111. Example 3: Factor the trinomials.b.

112. Your Turn 2: Factor the trinomials.
a b.

113. If three terms: Factoring Trinomials
Example 3: Factor completely. If not possible, write prime. a) b)

114. Your Turn 2: Factor completely. If not possible, write prime.d) e) f) g) h)

115. Warm-up/review Lesson 5.1- 5.4:
I. Factor completely (lesson 5.4):
1) 40xy + 30x – 100y – ) 3y2 + 21y +36
II. Simplify (lessons 5.1, 5.2 & 5.3):
–5(2x) ) x5 • x-12 • x
5) (15q6 + 5q2)(5q2)-1
III. Simplify by long division or synthetic: (it’s your choice!)
6) (x3 + 7x2 + 14x + 3) ÷ (x + 2)

116. Part II

117. Warm-up/review Lesson 5.1- 5.4:
I. Factor completely (lesson 5.4):
1) 40xy + 30x – 100y – ) 3y2 + 21y +36
5(4y + 3)(2x – 5) (y + 4)(y + 3)
II. Simplify (lessons 5.1, 5.2 & 5.3):
–5(2x) x ) x5 • x-12 • x x6
5) (15q6 + 5q2)(5q2) q4 + 1
III. Simplify by long division or synthetic: (it’s your choice!)
6) (x3 + 7x2 + 14x + 3) ÷ (x + 2)
x2 + 5x
x + 2

118. Factoring Polynomials lesson 5-4
Objectives:
Factor polynomials with three or two terms
Identify perfect square trinomials (PST), difference of two squares,
& sum or difference of cubes

119. Special Case of trinomial: PST Go to pg. 20
Example 4: a) b)

120. I) Difference of two squares Example 5:
If two terms:
I) Difference of two squares
Example 5: b) a) c) d)

121. If two terms:
II) Sum of two cubes
Example 6:

122. If two terms:
III) Difference of two cubes
Example 7:

123. Your Turn 3: Factor completely

124. Your Turn 3: Factor completely
Difference of two squares;
4(f + 4)(f – 4)
n3 – 125 c.
Difference of two cubes;
(n – 5)(n2 + 5n + 25)

125. Simplify quotients of polynomials by factoring
Assume no denominator is zero.
Example
Your Turn

126. Lesson 5-4: Factoring polynomials
Homework: 5.4 (worksheet)

127. Simplify. 1. 2. 3. 4. Factor the following 5. 6. 7.
Warm- up: Lesson 5.4
Alg. 2B
Simplify. 1. 2. 3. 4.
Synthetic division
Factor the following 5. 6. 7.