This document provides examples and explanations of identifying and simplifying polynomials. It begins with defining polynomials as expressions with variables that have non-negative integer exponents. Examples are provided of identifying polynomials and non-polynomials. Later pages discuss adding and subtracting polynomials by combining like terms. Worked examples are shown of simplifying polynomial expressions by removing parentheses and combining like terms. Degrees of polynomials are identified.

1. Identifying &
Simplifying Polynomials

2. What is a Polynomial?
View this Cool Math lesson introducing
Polynomials.
There is only the 1 page of notes.

3. Identify each of the following as
a Polynomial or Not a Polynomial
2
3x y

4. Identify each of the following as
a Polynomial or Not a Polynomial
2 Is a polynomial because the
3x y
variables have non-negative integer
exponents.

5. Identify each of the following as
a Polynomial or Not a Polynomial
2 Is a polynomial because the
3x y
variables have non-negative integer
exponents.
−6
5x

6. Identify each of the following as
a Polynomial or Not a Polynomial
2 Is a polynomial because the
3x y
variables have non-negative integer
exponents.
−6 Is a not a polynomial because
5x variable has a negative integer
exponents.

7. Identify each of the following as
a Polynomial or Not a Polynomial
2 Is a polynomial because the
3x y
variables have non-negative integer
exponents.
−6 Is a not a polynomial because
5x variable has a negative integer
exponents.
10
0.4xy − 5xy

8. Identify each of the following as
a Polynomial or Not a Polynomial
2 Is a polynomial because the
3x y
variables have non-negative integer
exponents.
−6 Is a not a polynomial because
5x variable has a negative integer
exponents.
Is a polynomial because the
10
0.4xy − 5xy variables have non-negative integer
exponents. The square root of a
number does not prevent an
expression from being a polynomial.

9. Describe Polynomials
View this Cool Math lesson introducing
terms describing Polynomials.
There are 3 pages of notes.
Be sure to complete the Your Turn problems
in your notebook. Check your answers on
the next slides.

10. Your Turn on Page 3
Find the degree of 3x - x2 .

11. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.

12. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .

13. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .
The degree is 5.

14. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .
The degree is 5.
Find the degree of 7x - 1.

15. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .
The degree is 5.
Find the degree of 7x - 1.
The degree is 1.

16. Adding & Subtracting
Polynomials
View this Cool Math lesson to learn how to
add and subtract polynomials.
There are 5 pages of notes.
Be sure to complete the Try It and Your
Turn problems in your notebook. Check
your answers on the next slides.

17. Try It on CoolMath Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )

18. Try It on CoolMath Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.

19. Try It on CoolMath Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z

20. Try It on CoolMath Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together.

21. Try It on CoolMath Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together. ( 3x + 6x ) + ( −4w + 2w ) + (1z + 5z )

22. Try It on CoolMath Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together. ( 3x + 6x ) + ( −4w + 2w ) + (1z + 5z )
Combine like terms.

23. Try It on CoolMath Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together. ( 3x + 6x ) + ( −4w + 2w ) + (1z + 5z )
Combine like terms.
9x − 2w + 6z

24. What is the degree of the
previous answer?
9x − 2w + 6z

25. What is the degree of the
previous answer?
9x − 2w + 6z
The first term is degree 1.

26. What is the degree of the
previous answer?
9x − 2w + 6z
The first term is degree 1.
Second term is degree 1.

27. What is the degree of the
previous answer?
9x − 2w + 6z
The first term is degree 1.
Second term is degree 1.
Third term is degree 1.

28. What is the degree of the
previous answer?
9x − 2w + 6z
The first term is degree 1.
Second term is degree 1.
Third term is degree 1.
Therefore, the degree of the polynomial is
1.

29. Your Turn on
CoolMath Page 3
Simplify. ( 8x 2
) ( 2
)
− x + 3 + 2x + 6x + 1

30. Your Turn on
CoolMath Page 3
Simplify. ( 8x 2
) ( 2
)
− x + 3 + 2x + 6x + 1
Remove parenthesis.

31. Your Turn on
CoolMath Page 3
Simplify. ( 8x 2
) ( 2
)
− x + 3 + 2x + 6x + 1
Remove parenthesis.
2 2
8x − x + 3 + 2x + 6x + 1

32. Your Turn on
CoolMath Page 3
Simplify. ( 8x 2
) ( 2
)
− x + 3 + 2x + 6x + 1
Remove parenthesis.
2 2
8x − x + 3 + 2x + 6x + 1
Rearrange so like
terms are together.

33. Your Turn on
CoolMath Page 3
Simplify. ( 8x 2
) ( 2
− x + 3 + 2x + 6x + 1 )
Remove parenthesis.
2 2
8x − x + 3 + 2x + 6x + 1
Rearrange so like
terms are together.
( 8x 2
)
+ 2x + ( −x + 6x ) + ( 3 + 1)
2

34. Your Turn on
CoolMath Page 3
Simplify. ( 8x 2
) ( 2
− x + 3 + 2x + 6x + 1 )
Remove parenthesis.
2 2
8x − x + 3 + 2x + 6x + 1
Rearrange so like
terms are together.
( 8x 2
)
+ 2x + ( −x + 6x ) + ( 3 + 1)
2
Combine like terms.

35. Your Turn on
CoolMath Page 3
Simplify. ( 8x 2
) ( 2
− x + 3 + 2x + 6x + 1 )
Remove parenthesis.
2 2
8x − x + 3 + 2x + 6x + 1
Rearrange so like
terms are together.
( 8x 2
)
+ 2x + ( −x + 6x ) + ( 3 + 1)
2
Combine like terms.
2
10x + 5x + 4

36. What is the degree of the
previous answer?
2
10x + 5x + 4

37. What is the degree of the
previous answer?
2
10x + 5x + 4
The first term is degree 2.

38. What is the degree of the
previous answer?
2
10x + 5x + 4
The first term is degree 2.
Second term is degree 1.

39. What is the degree of the
previous answer?
2
10x + 5x + 4
The first term is degree 2.
Second term is degree 1.
Third term is degree 0.

40. What is the degree of the
previous answer?
2
10x + 5x + 4
The first term is degree 2.
Second term is degree 1.
Third term is degree 0.
Therefore, the degree of the polynomial is
2.

41. Try It on CoolMath Page 3
Simplify. ( 4a b − ab + 3b
2 2
) ( 2 2
− 1 + a b + 8ab − 6a + 5 )

42. Try It on CoolMath Page 3
Simplify. ( 4a b − ab + 3b
2 2
) ( 2 2
− 1 + a b + 8ab − 6a + 5 )
Remove parenthesis.

43. Try It on CoolMath Page 3
Simplify. ( 4a b − ab + 3b
2 2
) ( 2 2
− 1 + a b + 8ab − 6a + 5 )
Remove parenthesis.
2 2 2 2
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5

44. Try It on CoolMath Page 3
Simplify. ( 4a b − ab + 3b
2 2
) ( 2 2
− 1 + a b + 8ab − 6a + 5 )
Remove parenthesis.
2 2 2 2
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
Rearrange so like terms are together.

45. Try It on CoolMath Page 3
Simplify. ( 4a b − ab + 3b
2 2
) ( 2
− 1 + a b + 8ab − 6a + 5 2
)
Remove parenthesis.
2 2 2 2
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
Rearrange so like terms are together.
( 4a b + a b ) + ( −ab + 8ab ) + 3b
2 2 2
− 6a + ( −1 + 5 )
2

46. Try It on CoolMath Page 3
Simplify. ( 4a b − ab + 3b
2 2
) ( 2
− 1 + a b + 8ab − 6a + 5 2
)
Remove parenthesis.
2 2 2 2
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
Rearrange so like terms are together.
( 4a b + a b ) + ( −ab + 8ab ) + 3b
2 2 2
− 6a + ( −1 + 5 )
2
Combine like terms.

47. Try It on CoolMath Page 3
Simplify. ( 4a b − ab + 3b
2 2
) ( 2
− 1 + a b + 8ab − 6a + 5 2
)
Remove parenthesis.
2 2 2 2
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
Rearrange so like terms are together.
( 4a b + a b ) + ( −ab + 8ab ) + 3b
2 2 2
− 6a + ( −1 + 5 )
2
Combine like terms.
2 2 2
5a b + 7ab + 3b − 6a + 4

48. What is the degree of the
previous answer?
2 2 2
5a b + 7ab + 3b − 6a + 4

49. What is the degree of the
previous answer?
2 2 2
5a b + 7ab + 3b − 6a + 4
The first term is degree 3.

50. What is the degree of the
previous answer?
2 2 2
5a b + 7ab + 3b − 6a + 4
The first term is degree 3.
Second term is degree 2.

51. What is the degree of the
previous answer?
2 2 2
5a b + 7ab + 3b − 6a + 4
The first term is degree 3.
Second term is degree 2.
Third term is degree 2.

52. What is the degree of the
previous answer?
2 2 2
5a b + 7ab + 3b − 6a + 4
The first term is degree 3.
Second term is degree 2.
Third term is degree 2.
Fourth term is degree 2.

53. What is the degree of the
previous answer?
2 2 2
5a b + 7ab + 3b − 6a + 4
The first term is degree 3.
Second term is degree 2.
Third term is degree 2.
Fourth term is degree 2.
Fifth term is degree 0

54. What is the degree of the
previous answer?
2 2 2
5a b + 7ab + 3b − 6a + 4
The first term is degree 3.
Second term is degree 2.
Third term is degree 2.
Fourth term is degree 2.
Fifth term is degree 0
Therefore, the degree of the polynomial is 3.

55. Your Turn on
CoolMath Page 4
Simplify. ( 3x − 4x + 8 ) − ( x
2 2
+ 7x − 5 )

56. Your Turn on
CoolMath Page 4
Simplify. ( 3x − 4x + 8 ) − ( x
2 2
+ 7x − 5 )
Remove parenthesis but
distribute the negative.

57. Your Turn on
CoolMath Page 4
Simplify. ( 3x − 4x + 8 ) − ( x
2 2
+ 7x − 5 )
Remove parenthesis but
distribute the negative.
2 2
3x − 4x + 8 − x − 7x + 5

58. Your Turn on
CoolMath Page 4
Simplify. ( 3x − 4x + 8 ) − ( x
2 2
+ 7x − 5 )
Remove parenthesis but
distribute the negative.
2 2
3x − 4x + 8 − x − 7x + 5
Rearrange so like
terms are together.

59. Your Turn on
CoolMath Page 4
Simplify. ( 3x − 4x + 8 ) − ( x
2 2
+ 7x − 5 )
Remove parenthesis but
distribute the negative.
2 2
3x − 4x + 8 − x − 7x + 5
Rearrange so like
terms are together.
( 3x 2
−x 2
) + ( −4x − 7x ) + ( 8 + 5 )

60. Your Turn on
CoolMath Page 4
Simplify. ( 3x − 4x + 8 ) − ( x
2 2
+ 7x − 5 )
Remove parenthesis but
distribute the negative.
2 2
3x − 4x + 8 − x − 7x + 5
Rearrange so like
terms are together.
( 3x 2
−x 2
) + ( −4x − 7x ) + ( 8 + 5 )
Combine like terms.

61. Your Turn on
CoolMath Page 4
Simplify. ( 3x − 4x + 8 ) − ( x
2 2
+ 7x − 5 )
Remove parenthesis but
distribute the negative.
2 2
3x − 4x + 8 − x − 7x + 5
Rearrange so like
terms are together.
( 3x 2
−x 2
) + ( −4x − 7x ) + ( 8 + 5 )
Combine like terms.
2
2x − 11x + 13

62. What is the degree of the
previous answer?
2
2x − 11x + 13

63. What is the degree of the
previous answer?
2
2x − 11x + 13
The first term is degree 2.

64. What is the degree of the
previous answer?
2
2x − 11x + 13
The first term is degree 2.
Second term is degree 1.

65. What is the degree of the
previous answer?
2
2x − 11x + 13
The first term is degree 2.
Second term is degree 1.
Third term is degree 0.

66. What is the degree of the
previous answer?
2
2x − 11x + 13
The first term is degree 2.
Second term is degree 1.
Third term is degree 0.
Therefore, the degree of the polynomial is
2.

67. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .

68. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.

69. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.
(x 2 2
) ( 2
− 4xy + 5y − 3x − 6xy + y 2
)

70. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.
(x 2 2
) ( 2
− 4xy + 5y − 3x − 6xy + y 2
)
Remove parenthesis but
distribute the negative.

71. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.
(x 2 2
) ( 2
− 4xy + 5y − 3x − 6xy + y 2
)
Remove parenthesis but
distribute the negative. 2 2 2 2
x − 4xy + 5y − 3x + 6xy − y

72. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.
(x 2 2
) ( 2
− 4xy + 5y − 3x − 6xy + y 2
)
Remove parenthesis but
distribute the negative. 2 2 2 2
x − 4xy + 5y − 3x + 6xy − y
Rearrange so like
terms are together.

73. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.
(x 2 2
) ( 2
− 4xy + 5y − 3x − 6xy + y 2
)
Remove parenthesis but
distribute the negative. 2 2 2 2
x − 4xy + 5y − 3x + 6xy − y
Rearrange so like
terms are together.
(x 2
)
− 3x + ( −4xy + 6xy ) + 5y − y
2
( 2 2
)

74. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.
(x 2 2
) ( 2
− 4xy + 5y − 3x − 6xy + y 2
)
Remove parenthesis but
distribute the negative. 2 2 2 2
x − 4xy + 5y − 3x + 6xy − y
Rearrange so like
terms are together.
(x 2
)
− 3x + ( −4xy + 6xy ) + 5y − y
2
( 2 2
)
Combine like terms.

75. Your Turn on
CoolMath Page 5
2 2 2 2
Subtract 3x − 6xy + y from x − 4xy + 5y .
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y 2 2
)
Remove parenthesis but
distribute the negative. 2 2 2 2
x − 4xy + 5y − 3x + 6xy − y
Rearrange so like
terms are together.
(x 2
)
− 3x + ( −4xy + 6xy ) + 5y − y
2
( 2 2
)
Combine like terms. 2 2
−2x + 2xy + 4y

76. What is the degree of the
previous answer?
2 2
−2x + 2xy + 4y

77. What is the degree of the
previous answer?
2 2
−2x + 2xy + 4y
The first term is degree 2.

78. What is the degree of the
previous answer?
2 2
−2x + 2xy + 4y
The first term is degree 2.
Second term is degree 2.

79. What is the degree of the
previous answer?
2 2
−2x + 2xy + 4y
The first term is degree 2.
Second term is degree 2.
Third term is degree 2.

80. What is the degree of the
previous answer?
2 2
−2x + 2xy + 4y
The first term is degree 2.
Second term is degree 2.
Third term is degree 2.
Therefore, the degree of the polynomial is
2.

81. Adding & Subtracting
Polynomials Practice
Practice Adding & Subtracting Polynomials at this Cool
Math website.
Select the “Give me a Problem” button to keep trying
problems.
Do your work in a notebook.
When you select “What’s the Answer?” your answer is
erased and correct answer is displayed. Having your
work in a notebook will allow you to compare your
answer to the correct answer.
Keep working problems until you get 4 out of 5 correct.

82. Way to Go!
You’ve finished reviewing
the notes for Identifying,
Adding and Subtracting
Polynomials.
Proceed to the Assignment.