1. ALGEBRA 1 UNIT 8 POLYNOMIAL EXPRESSIONS AND FACTORING
Unit Essential Questions
Are two algebraic expressions that appear to be different actually equivalent?
What is the relationship between properties of real numbers and properties of polynomials?

2. ADDING AND SUBTRACTING POLYNOMIALS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.

3. DO NOW
Simplify. 1) 2) 3)

4. KEY CONCEPTS AND VOCABULARY
A monomial is a real number, a variable, or the product of real numbers and variables (Note: the variables must have positive integer exponents to be a monomial). The degree of a monomial is the sum of the exponents of its variables. A polynomial is a monomial or a sum of monomials. Standard form of a polynomial means that the degrees of its monomial terms are written in descending order. The degree of a polynomial is the same as the degree of the monomial with the greatest exponent.

5. KEY CONCEPTS AND VOCABULARY
CLASSIFICATION OF POLYNOMIALS
DEGREE
NUMBER OF TERMS
Constant 1
Monomial
Linear 2
Binomial
Quadratic 3
Trinomial
Cubic 4
Polynomial with 4 terms
EXAMPLES OF
MONOMIALS
EXAMPLES OF NOT 6 g

6. EXAMPLE 1: IDENTIFYING POLYNOMIALS
Determine whether each expression is a polynomial. If it is a polynomial, classify the polynomial by the degree and number of terms. a) b) c) d) 5

7. EXAMPLE 2: WRITING POLYNOMIALS IN STANDARD FORM
Write the polynomial in standard form. Then identify the leading coefficient. a) b) c)

8. EXAMPLE 3: ADDING POLYNOMIALS
Simplify. a) b)

9. EXAMPLE 4: SUBTRACTING POLYNOMIALS
Simplify. a) b)

10. EXAMPLE 5: SIMPLIFYING USING GEOMETRIC FORMULAS
Express the perimeter as a polynomial. a) b)

11. EXAMPLE 6: ADDING AND SUBTRACTING POLYNOMIALS IN REAL-WORLD APPLICATIONS
The equations and represent the number of Miami Heat hats, H, and the number of Cleveland Cavalier hats, C, sold in m months at a sports store.
Write an equation for the total, T, of Heat and Cavalier hats sold.
Predict the number of Heat and Cavalier hats sold in 9 months.

12. RATE YOUR UNDERSTANDING
ADDING AND SUBTRACTING POLYNOMIALS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING
LEARNING SCALE 4
I am able to
add and subtract polynomials in real-world applications or in more challenging problems that I have never previously attempted 3
identify a polynomial and write polynomials in standard form
add and subtract polynomials 2
identify a polynomial and write polynomials in standard form with help
add and subtract polynomials with help 1
identify the degree of a monomial
TARGET

13. MULTIPLYING A POLYNOMIAL BY A MONOMIAL
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

14. WARM UP
Simplify. 1) 2) 3)

15. KEY CONCEPTS AND VOCABULARY
You can use the Distribution Property to multiply a monomial by a polynomial.

16. EXAMPLE 1: MULTIPLYING A POLYNOMIAL BY A MONOMIAL
Simplify. a) b)

17. EXAMPLE 2: SIMPLIFYING EXPRESSIONS WITH A PRODUCT OF A POLYNOMIAL AND A MONOMIAL
Simplify. a) b)

18. EXAMPLE 3: SIMPLIFYING USING GEOMETRIC FORMULAS
Express the area as a polynomial. a) b)

19. EXAMPLE 4: SOLVING EQUATIONS WITH POLYNOMIALS ON EACH SIDE
Solve. a) b)

20. RATE YOUR UNDERSTANDING
MULTIPLYING A POLYNOMIAL BY A MONOMIAL
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
RATING
LEARNING SCALE 4
I am able to
multiply a polynomial by a monomial in more challenging problems that I have never previously attempted (such as solving equations) 3
multiply a polynomial by a monomial 2
multiply a polynomial by a monomial with help 1
understand that the distributive property can be applied to polynomials
TARGET

21. MULTIPLYING POLYNOMIALS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

22. WARM UP
Simplify. 1) 2) 3)

23. KEY CONCEPTS AND VOCABULARY
METHODS FOR MULTIPLYING POLYNOMIALS
DISTRIBUTIVE PROPERTY METHOD
FOIL METHOD
Example:

24. EXAMPLE 1: FINDING THE PRODUCT OF TWO BINOMIALS USING THE DISTRIBUTIVE PROPERTY
Simplify using the distributive property. a) b) c)

25. EXAMPLE 2: FINDING THE PRODUCT OF TWO BINOMIALS USING THE FOIL METHOD
Simplify using the FOIL method. a) b) c)

26. EXAMPLE 3: FINDING THE PRODUCT OF A BINOMIAL AND TRINOMIAL
Simplify using the distributive property. a) b) c)

27. EXAMPLE 4: SIMPLIFYING PRODUCTS
Simplify. a) b)

28. RATE YOUR UNDERSTANDING
MULTIPLYING POLYNOMIALS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
RATING
LEARNING SCALE 4
I am able to
multiply two binomials or a binomial by a trinomial in more challenging problems that I have never previously attempted 3
multiply two binomials or a binomial by a trinomial 2
multiply two binomials or a binomial by a trinomial with help 1
understand the distributive property
TARGET

29. SPECIAL PRODUCTS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

30. WARM UP
Simplify. 1) 2) 3)

31. KEY CONCEPTS AND VOCABULARY
MULTIPLYING SPECIAL CASES
THE SQUARE OF A BINOMIAL
THE PRODUCT OF A SUM AND DIFFERENCE
(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2 Or
(a – b)2 = (a – b)(a – b) = a2 – 2ab + b2
(a + b)(a – b) = a2 – b2

32. EXAMPLE 1: SIMPLIFYING THE SQUARE OF A BINOMIAL (SUM)
Simplify. a) b) c)

33. EXAMPLE 2: SIMPLIFYING THE SQUARE OF A BINOMIAL (DIFFERENCE)
Simplify. a) b) c)

34. EXAMPLE 3: SIMPLIFYING THE PRODUCT OF A SUM AND DIFFERENCE
Simplify. a) b) c) d)

35. EXAMPLE 4: SIMPLIFYING MORE CHALLENGING PROBLEMS WITH SPECIAL CASES
Simplify. a) b) c) d)

36. EXAMPLE 4: SIMPLIFYING MORE CHALLENGING PROBLEMS WITH SPECIAL CASES
Simplify. e) f)

37. RATE YOUR UNDERSTANDING
SPECIAL PRODUCTS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
RATING
LEARNING SCALE 4
I am able to
simplify special products in more challenging problems that I have never previously attempted 3
find the square of a binomial
find the product of a sum and difference 2
find the square of a binomial with help
find the product of a sum and difference with help 1
understand that there are special rules to simplify the square of a binomial and the product of a sum and difference
TARGET

38. FUNCTION OPERATIONS
MACC.912.F-BF.A.1b: Combine standard function types using arithmetic operations.

39. WARM UP Perform the indicated operation. (x + 2x2 – 4) + (x2 – 3x + 9)

40. KEY CONCEPTS AND VOCABULARY
FUNCTION OPERATIONS
ADDITION
(f + g)(x) = f(x) + g(x)
SUBTRACTION
(f – g)(x) = f(x) – g(x)
MULTIPLICATION
(f • g)(x) = f(x) • g(x)
DIVISION
(f / g)(x) = f(x) / g(x), g(x) ≠ 0

41. EXAMPLE 1: FUNCTION OPERATIONS WITH LINEAR FUNCTIONS
Let f(x) = –2x + 6 and g(x) = 5x – 7. Use the functions f(x) and g(x) to find produce a new function h(x). a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x) c) h(x) = (f • g)(x) d) h(x) = (f / g)(x)

42. EXAMPLE 2: FUNCTION OPERATIONS WITH LINEAR AND QUADRATIC FUNCTIONS
Let and . Use the functions f(x) and g(x) to find produce a new function h(x). a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x) c) h(x) = (f • g)(x) d) h(x) = (f / g)(x)

43. EXAMPLE 3: FUNCTION OPERATIONS WITH LINEAR AND EXPONENTIAL FUNCTIONS
Let and . Use the functions f(x) and g(x) to find produce a new function h(x). a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x) c) h(x) = (f • g)(x) d) h(x) = (f / g)(x)

44. EXAMPLE 4: FUNCTION OPERATIONS WITH QUADRATIC AND EXPONENTIAL FUNCTIONS
Let and . Use the functions f(x) and g(x) to find produce a new function h(x). a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x)

45. RATE YOUR UNDERSTANDING
FUNCTION OPERATIONS
MACC.912.F-BF.A.1b: Combine standard function types using arithmetic operations.
RATING
LEARNING SCALE 4
I am able to
perform arithmetic operations with functions in more challenging problems that I have never previously attempted 3
perform arithmetic operations with functions 2
perform arithmetic operations with functions with help 1
understand that you can add, subtract, multiply, and divide functions
TARGET

46. GREATEST COMMON FACTOR
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.

47. WARM UP
Multiply. 1) 3(x – 2) 2) x(x – 9) 3) (x + 5)(x – 9) 4) x2(x2 – 4x + 5)

48. KEY CONCEPTS AND VOCABULARY
You can work backwards to express a polynomial as the product of polynomials. Factoring – rewriting an expression as the product of polynomials. (un-distributing) Greatest Common Factor – the largest quantity that is a factor of all the integers or polynomials involved.

49. EXAMPLE 1: FINDING THE GREATEST COMMON FACTOR FROM A LIST OF INTEGERS
Find the greatest common factor of each list of numbers.
12 and 8
7 and 20
4, 12, and 26

50. EXAMPLE 2: FINDING THE GREATEST COMMON FACTOR FROM A LIST OF MONOMIALS
Find the greatest common factor of each list of monomials.
a) x3 and x7
6x5 and 4x3
c) 3xy2 and 12x2y

51. EXAMPLE 3: FACTORING THE GREATEST COMMON FACTOR
Factor the greatest common factor in each of the following polynomials.
15x b) 8m2 + 4m
c) 3x2 + 6x d) 5x2 + 13y

52. EXAMPLE 3: FACTORING THE GREATEST COMMON FACTOR
Factor the greatest common factor in each of the following polynomials.
6x3 – 9x2 + 12x f) 14x3y + 7x2y – 7xy  
g) 6(x + 2) – y(x + 2) h) xy(y + 1) – (y + 1)

53. EXAMPLE 4: SIMPLIFYING USING GEOMETRIC FORMULAS
Express the perimeter in factored form. a) b)

54. RATE YOUR UNDERSTANDING
GREATEST COMMON FACTOR
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
RATING
LEARNING SCALE 4
I am able to
rewrite an expression as the product of the greatest common factor and the remaining polynomial in more challenging problems that I have never previously attempted 3
rewrite an expression as the product of the greatest common factor and the remaining polynomial 2
rewrite an expression as the product of the greatest common factor and the remaining polynomial with help 1
identify the greatest common factor
TARGET

55. FACTORING BY GROUPING
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1b: Interpret complicated expressions by viewing one or more of their parts as a single entity.

56. WARM UP Factor out the greatest common factor. 1) x(x + 2) – 3(x + 2)
3) 4x(y + 12) + (y + 12)

57. KEY CONCEPTS AND VOCABULARY
Factor by Grouping – factor a polynomial by grouping the terms of the polynomial and looking for common factors. FACTORING BY GROUPING (4 TERMS) ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

58. EXAMPLE 1: FACTORING A POLYNOMIAL BY GROUPING
a) x3 + 2x2 – 3x – 6
b) x3 + 4x + x2 + 4
c) 2x3 – x2 – 10x + 5

59. EXAMPLE 1: FACTORING A POLYNOMIAL BY GROUPING
Factor. d) ab + 2a + 8b + 16 e) xy – 6x + 6y – 36 f) 9rs – 45r – 7s + 35

60. RATE YOUR UNDERSTANDING
FACTORING BY GROUPING
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1b: Interpret complicated expressions by viewing one or more of their parts as a single entity.
RATING
LEARNING SCALE 4
I am able to
factor polynomials by grouping in more challenging problems that I have never previously attempted 3
factor polynomials by grouping 2
factor polynomials by grouping with help 1
understand that I can group polynomials to factor
TARGET

61. FACTORING X2 +BX + C
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.

62. WARM UP
Multiply.
(x + 2)(x – 5)
(y – 7)(x – 1)
(x + y)(2x – y)

63. KEY CONCEPTS AND VOCABULARY
STEPS FOR FACTORING TRINOMIALS WITH LEADING COEFFICIENT = 1 (X2 + BX + C)
Find two integers that multiply to c and add to b.
Write the binomial factors as (x + ___)(x + ___) filling in the blank with the two integers found
Check your answer by using the distributive property or the FOIL method
Example:

64. EXAMPLE 1: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B AND C ARE POSITIVE
Factor. a) b) c)

65. EXAMPLE 2: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B IS NEGATIVE AND C IS POSITIVE
Factor. a) b) c)

66. EXAMPLE 3: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B IS POSITIVE AND C IS NEGATIVE
Factor. a) b) c)

67. EXAMPLE 4: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B AND C ARE NEGATIVE
Factor. a) b) c)

68. EXAMPLE 5: FACTORING TRINOMIALS IN THE FORM X2 + BX + C AFTER FACTORING OUT A GCF
Factor. a) b) c)

69. EXAMPLE 6: APPLYING FACTORING TRINOMIALS TO GEOMETRIC FORMULAS
The area of a rectangle is given by the trinomial
What are the possible dimensions of the rectangle?
What are the exact dimensions if the width of the rectangle is 3 inches?

70. RATE YOUR UNDERSTANDING
FACTORING X2 +BX + C
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING
LEARNING SCALE 4
I am able to
factor trinomials of the form x2 + bx + c in more challenging problems that I have never previously attempted 3
factor trinomials of the form x2 + bx + c 2
factor trinomials of the form x2 + bx + c with help 1
understand that some trinomials can be written as the product of two binomials
TARGET

71. FACTORING AX2 +BX + C
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.

72. WARM UP
Write 2 different expressions that have a factor of (x + 6).

73. KEY CONCEPTS AND VOCABULARY
Steps for Factoring Trinomials with Leading Coefficient ≠ 1 (ax2 + bx + c)
Find two integers that multiply to ac and add to b.
Rewrite the trinomial by splitting the middle term (b term) into the two integers found.
Factoring by grouping
Check your answer by using the distributive property or the FOIL method
Example:

74. EXAMPLE 1: FACTORING TRINOMIALS IN THE FORM AX2 + BX + C
Factor. a) b) c)

75. RATE YOUR UNDERSTANDING
FACTORING AX2 +BX + C
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING
LEARNING SCALE 4
I am able to
factor trinomials of the form ax2 + bx + c in more challenging problems that I have never previously attempted 3
factor trinomials of the form ax2 + bx + c 2
factor trinomials of the form ax2 + bx + c with help 1
understand that some trinomials can be written as the product of two binomials
TARGET

76. FACTORING SPECIAL CASES
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.

77. WARM UP
Multiply. 1) (2x – 7)(2x – 7) 2) (4x + 3)(4x – 3)

78. KEY CONCEPTS AND VOCABULARY
FACTORING SPECIAL CASES
PERFECT SQUARE TRINOMIAL
DIFFERENCE OF TWO SQUARES
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2 Or
a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2
a2 – b2 = (a + b)(a – b)

79. EXAMPLE 1: FACTORING PERFECT SQUARE TRINOMIALS
Factor. a) b) c)

80. EXAMPLE 2: FACTORING DIFFERENCE OF TWO SQUARES
Factor. a) b) c)

81. RATE YOUR UNDERSTANDING
FACTORING SPECIAL CASES
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING
LEARNING SCALE 4
I am able to
factor perfect-square trinomials and the differences of two squares in more challenging problems that I have never previously attempted 3
factor perfect-square trinomials and the differences of two squares 2
factor perfect-square trinomials and the differences of two squares with help 1
understand that some polynomials can be written as the product of two binomials
TARGET