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

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Transcript

  • 1. Identifying &Simplifying Polynomials
  • 2. What is a Polynomial?View this Cool Math lesson introducingPolynomials.There is only the 1 page of notes.
  • 3. Identify each of the following asa Polynomial or Not a Polynomial 23x y
  • 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. 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. 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. 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. 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. 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. Your Turn on Page 3Find the degree of 3x - x2 .
  • 11. Your Turn on Page 3Find the degree of 3x - x2 . The degree is 2.
  • 12. Your Turn on Page 3Find the degree of 3x - x2 . The degree is 2.Find the degree of a 4b - 7a 2b2 .
  • 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. 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. 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. 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. Try It on CoolMath Page 2Simplify: ( 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 + 6zThe first term is degree 1.
  • 26. What is the degree of the previous answer? 9x − 2w + 6zThe first term is degree 1.Second term is degree 1.
  • 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. 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. Your Turn on CoolMath Page 3Simplify. ( 8x 2 ) ( 2 ) − x + 3 + 2x + 6x + 1
  • 30. Your Turn on CoolMath Page 3 Simplify. ( 8x 2 ) ( 2 ) − x + 3 + 2x + 6x + 1Remove parenthesis.
  • 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. 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. 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. 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. 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. What is the degree of the previous answer? 2 10x + 5x + 4
  • 37. What is the degree of the previous answer? 2 10x + 5x + 4The first term is degree 2.
  • 38. What is the degree of the previous answer? 2 10x + 5x + 4The first term is degree 2.Second term is degree 1.
  • 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. 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. Try It on CoolMath Page 3Simplify. ( 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 + 5Rearrange 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 + 5Rearrange 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 + 5Rearrange so like terms are together. ( 4a b + a b ) + ( −ab + 8ab ) + 3b 2 2 2 − 6a + ( −1 + 5 ) 2Combine 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 + 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. 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 + 4The first term is degree 3.
  • 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. 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. 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. 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. 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. Your Turn on CoolMath Page 4Simplify. ( 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 butdistribute the negative.
  • 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. 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. 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. 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. 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. What is the degree of the previous answer? 2 2x − 11x + 13
  • 63. What is the degree of the previous answer? 2 2x − 11x + 13The first term is degree 2.
  • 64. What is the degree of the previous answer? 2 2x − 11x + 13The first term is degree 2.Second term is degree 1.
  • 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. 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. Your Turn on CoolMath Page 5 2 2 2 2Subtract 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 butdistribute 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 butdistribute 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 butdistribute 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 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. 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. 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. 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 + 4yThe first term is degree 2.
  • 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. 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. 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. 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. Way to Go!You’ve finished reviewingthe notes for Identifying,Adding and SubtractingPolynomials.Proceed to the Assignment.