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# Notes 12.1 identifying, adding & subtracting polynomials

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• ### Notes 12.1 identifying, adding & subtracting polynomials

1. 1. Identifying &Simplifying Polynomials
2. 2. What is a Polynomial?View this Cool Math lesson introducingPolynomials.There is only the 1 page of notes.
3. 3. Identify each of the following asa Polynomial or Not a Polynomial 23x y
4. 4. Identify each of the following asa Polynomial or Not a Polynomial 2 Is a polynomial because the3x y variables have non-negative integer exponents.
5. 5. Identify each of the following asa Polynomial or Not a Polynomial 2 Is a polynomial because the3x y variables have non-negative integer exponents. −65x
6. 6. Identify each of the following asa Polynomial or Not a Polynomial 2 Is a polynomial because the3x y variables have non-negative integer exponents. −6 Is a not a polynomial because5x variable has a negative integer exponents.
7. 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. 100.4xy − 5xy
8. 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 100.4xy − 5xy variables have non-negative integer exponents. The square root of a number does not prevent an expression from being a polynomial.
9. 9. Describe PolynomialsView this Cool Math lesson introducingterms describing Polynomials.There are 3 pages of notes.Be sure to complete the Your Turn problemsin your notebook. Check your answers onthe next slides.
10. 10. Your Turn on Page 3Find the degree of 3x - x2 .
11. 11. Your Turn on Page 3Find the degree of 3x - x2 . The degree is 2.
12. 12. Your Turn on Page 3Find the degree of 3x - x2 . The degree is 2.Find the degree of a 4b - 7a 2b2 .
13. 13. Your Turn on Page 3Find the degree of 3x - x2 . The degree is 2.Find the degree of a 4b - 7a 2b2 . The degree is 5.
14. 14. Your Turn on Page 3Find 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. 15. Your Turn on Page 3Find 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. 16. Adding & Subtracting PolynomialsView this Cool Math lesson to learn how toadd and subtract polynomials.There are 5 pages of notes.Be sure to complete the Try It and YourTurn problems in your notebook. Checkyour answers on the next slides.
17. 17. Try It on CoolMath Page 2Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
18. 18. Try It on CoolMath Page 2 Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )Remove parenthesis.
19. 19. Try It on CoolMath Page 2 Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )Remove parenthesis. 3x − 4w + z + 6x + 2w + 5z
20. 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. 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. 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. 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. 24. What is the degree of the previous answer? 9x − 2w + 6z
25. 25. What is the degree of the previous answer? 9x − 2w + 6zThe first term is degree 1.
26. 26. What is the degree of the previous answer? 9x − 2w + 6zThe first term is degree 1.Second term is degree 1.
27. 27. What is the degree of the previous answer? 9x − 2w + 6zThe first term is degree 1.Second term is degree 1.Third term is degree 1.
28. 28. What is the degree of the previous answer? 9x − 2w + 6zThe first term is degree 1.Second term is degree 1.Third term is degree 1.Therefore, the degree of the polynomial is1.
29. 29. Your Turn on CoolMath Page 3Simplify. ( 8x 2 ) ( 2 ) − x + 3 + 2x + 6x + 1
30. 30. Your Turn on CoolMath Page 3 Simplify. ( 8x 2 ) ( 2 ) − x + 3 + 2x + 6x + 1Remove parenthesis.
31. 31. Your Turn on CoolMath Page 3 Simplify. ( 8x 2 ) ( 2 ) − x + 3 + 2x + 6x + 1Remove parenthesis. 2 2 8x − x + 3 + 2x + 6x + 1
32. 32. Your Turn on CoolMath Page 3 Simplify. ( 8x 2 ) ( 2 ) − x + 3 + 2x + 6x + 1Remove parenthesis. 2 2 8x − x + 3 + 2x + 6x + 1 Rearrange so liketerms are together.
33. 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 liketerms are together. ( 8x 2 ) + 2x + ( −x + 6x ) + ( 3 + 1) 2
34. 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 liketerms are together. ( 8x 2 ) + 2x + ( −x + 6x ) + ( 3 + 1) 2Combine like terms.
35. 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 liketerms are together. ( 8x 2 ) + 2x + ( −x + 6x ) + ( 3 + 1) 2Combine like terms. 2 10x + 5x + 4
36. 36. What is the degree of the previous answer? 2 10x + 5x + 4
37. 37. What is the degree of the previous answer? 2 10x + 5x + 4The first term is degree 2.
38. 38. What is the degree of the previous answer? 2 10x + 5x + 4The first term is degree 2.Second term is degree 1.
39. 39. What is the degree of the previous answer? 2 10x + 5x + 4The first term is degree 2.Second term is degree 1.Third term is degree 0.
40. 40. What is the degree of the previous answer? 2 10x + 5x + 4The first term is degree 2.Second term is degree 1.Third term is degree 0.Therefore, the degree of the polynomial is2.
41. 41. Try It on CoolMath Page 3Simplify. ( 4a b − ab + 3b 2 2 ) ( 2 2 − 1 + a b + 8ab − 6a + 5 )
42. 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. 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. 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 + 5Rearrange so like terms are together.
45. 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 + 5Rearrange so like terms are together. ( 4a b + a b ) + ( −ab + 8ab ) + 3b 2 2 2 − 6a + ( −1 + 5 ) 2
46. 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 + 5Rearrange so like terms are together. ( 4a b + a b ) + ( −ab + 8ab ) + 3b 2 2 2 − 6a + ( −1 + 5 ) 2Combine like terms.
47. 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 + 5Rearrange so like terms are together. ( 4a b + a b ) + ( −ab + 8ab ) + 3b 2 2 2 − 6a + ( −1 + 5 ) 2Combine like terms. 2 2 2 5a b + 7ab + 3b − 6a + 4
48. 48. What is the degree of the previous answer? 2 2 2 5a b + 7ab + 3b − 6a + 4
49. 49. What is the degree of the previous answer? 2 2 2 5a b + 7ab + 3b − 6a + 4The first term is degree 3.
50. 50. What is the degree of the previous answer? 2 2 2 5a b + 7ab + 3b − 6a + 4The first term is degree 3.Second term is degree 2.
51. 51. What is the degree of the previous answer? 2 2 2 5a b + 7ab + 3b − 6a + 4The first term is degree 3.Second term is degree 2.Third term is degree 2.
52. 52. What is the degree of the previous answer? 2 2 2 5a b + 7ab + 3b − 6a + 4The first term is degree 3.Second term is degree 2.Third term is degree 2.Fourth term is degree 2.
53. 53. What is the degree of the previous answer? 2 2 2 5a b + 7ab + 3b − 6a + 4The 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. 54. What is the degree of the previous answer? 2 2 2 5a b + 7ab + 3b − 6a + 4The first term is degree 3.Second term is degree 2.Third term is degree 2.Fourth term is degree 2.Fifth term is degree 0Therefore, the degree of the polynomial is 3.
55. 55. Your Turn on CoolMath Page 4Simplify. ( 3x − 4x + 8 ) − ( x 2 2 + 7x − 5 )
56. 56. Your Turn on CoolMath Page 4 Simplify. ( 3x − 4x + 8 ) − ( x 2 2 + 7x − 5 )Remove parenthesis butdistribute the negative.
57. 57. Your Turn on CoolMath Page 4 Simplify. ( 3x − 4x + 8 ) − ( x 2 2 + 7x − 5 )Remove parenthesis butdistribute the negative. 2 2 3x − 4x + 8 − x − 7x + 5
58. 58. Your Turn on CoolMath Page 4 Simplify. ( 3x − 4x + 8 ) − ( x 2 2 + 7x − 5 )Remove parenthesis butdistribute the negative. 2 2 3x − 4x + 8 − x − 7x + 5 Rearrange so liketerms are together.
59. 59. Your Turn on CoolMath Page 4 Simplify. ( 3x − 4x + 8 ) − ( x 2 2 + 7x − 5 )Remove parenthesis butdistribute the negative. 2 2 3x − 4x + 8 − x − 7x + 5 Rearrange so liketerms are together. ( 3x 2 −x 2 ) + ( −4x − 7x ) + ( 8 + 5 )
60. 60. Your Turn on CoolMath Page 4 Simplify. ( 3x − 4x + 8 ) − ( x 2 2 + 7x − 5 )Remove parenthesis butdistribute the negative. 2 2 3x − 4x + 8 − x − 7x + 5 Rearrange so liketerms are together. ( 3x 2 −x 2 ) + ( −4x − 7x ) + ( 8 + 5 )Combine like terms.
61. 61. Your Turn on CoolMath Page 4 Simplify. ( 3x − 4x + 8 ) − ( x 2 2 + 7x − 5 )Remove parenthesis butdistribute the negative. 2 2 3x − 4x + 8 − x − 7x + 5 Rearrange so liketerms are together. ( 3x 2 −x 2 ) + ( −4x − 7x ) + ( 8 + 5 )Combine like terms. 2 2x − 11x + 13
62. 62. What is the degree of the previous answer? 2 2x − 11x + 13
63. 63. What is the degree of the previous answer? 2 2x − 11x + 13The first term is degree 2.
64. 64. What is the degree of the previous answer? 2 2x − 11x + 13The first term is degree 2.Second term is degree 1.
65. 65. What is the degree of the previous answer? 2 2x − 11x + 13The first term is degree 2.Second term is degree 1.Third term is degree 0.
66. 66. What is the degree of the previous answer? 2 2x − 11x + 13The first term is degree 2.Second term is degree 1.Third term is degree 0.Therefore, the degree of the polynomial is2.
67. 67. Your Turn on CoolMath Page 5 2 2 2 2Subtract 3x − 6xy + y from x − 4xy + 5y .
68. 68. Your Turn on CoolMath Page 5 2 2 2 2 Subtract 3x − 6xy + y from x − 4xy + 5y .Write as a subtraction.
69. 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. 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 butdistribute the negative.
71. 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 butdistribute the negative. 2 2 2 2 x − 4xy + 5y − 3x + 6xy − y
72. 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 butdistribute the negative. 2 2 2 2 x − 4xy + 5y − 3x + 6xy − y Rearrange so like terms are together.
73. 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 butdistribute 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. 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 butdistribute 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. 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 butdistribute 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. 76. What is the degree of the previous answer? 2 2 −2x + 2xy + 4y
77. 77. What is the degree of the previous answer? 2 2 −2x + 2xy + 4yThe first term is degree 2.
78. 78. What is the degree of the previous answer? 2 2 −2x + 2xy + 4yThe first term is degree 2.Second term is degree 2.
79. 79. What is the degree of the previous answer? 2 2 −2x + 2xy + 4yThe first term is degree 2.Second term is degree 2.Third term is degree 2.
80. 80. What is the degree of the previous answer? 2 2 −2x + 2xy + 4yThe first term is degree 2.Second term is degree 2.Third term is degree 2.Therefore, the degree of the polynomial is2.
81. 81. Adding & Subtracting Polynomials PracticePractice Adding & Subtracting Polynomials at this CoolMath website.Select the “Give me a Problem” button to keep tryingproblems.Do your work in a notebook.When you select “What’s the Answer?” your answer iserased and correct answer is displayed. Having yourwork in a notebook will allow you to compare youranswer to the correct answer.Keep working problems until you get 4 out of 5 correct.
82. 82. Way to Go!You’ve finished reviewingthe notes for Identifying,Adding and SubtractingPolynomials.Proceed to the Assignment.