1. Lecture 281 Unit 4 Lecture 28 Introduction to Quadratics Introduction to Quadratics

2. Lecture 282 Objectives Evaluate quadratic expressionsEvaluate quadratic expressions Identify the degree of a polynomialIdentify the degree of a polynomial Determine the number of terms in a polynomialDetermine the number of terms in a polynomial Add and subtract polynomialsAdd and subtract polynomials Multiply binomials using FOILMultiply binomials using FOIL

3. Lecture 283 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h = -16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 5 8 10

4. Lecture 284 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = 5 8 10

5. Lecture 285 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = -64+320 = 256 5 8 10

6. Lecture 286 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = -64+320 = 256 5 -16(5) 2 + 160(5) = 8 10

7. Lecture 287 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = -64+320 = 256 5 -16(5) 2 + 160(5) = -400+800 = 400 8 10

8. Lecture 288 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = -64+320 = 256 5 -16(5) 2 + 160(5) = -400+800 = 400 8 -16(8) 2 + 160(8) = 10

9. Lecture 289 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = -64+320 = 256 5 -16(5) 2 + 160(5) = -400+800 = 400 8 -16(8) 2 + 160(8) = -1024+1280 = 256 10

10. Lecture 2810 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = -64+320 = 256 5 -16(5) 2 + 160(5) = -400+800 = 400 8 -16(8) 2 + 160(8) = -1024+1280 = 256 10 -16(10) 2 + 160(10) =

11. Lecture 2811 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t 2 + 160 t 2 -16(2) 2 + 160(2) = -64+320 = 256 5 -16(5) 2 + 160(5) = -400+800 = 400 8 -16(8) 2 + 160(8) = -1024+1280 = 256 10 -16(10) 2 + 160(10) = -1600+1600 = 0

12. Lecture 2812 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. When will the arrow reach its maximum height? th 2256 5400 8256 100

13. Lecture 2813 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. When will the arrow reach its maximum height? th 2256 5400 8256 100 When time = 5 sec, height = 400 feet.

14. Lecture 2814 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. When will the arrow hit the ground? th 2256 5400 8256 100

15. Lecture 2815 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t 2 + 160 t t is the number of seconds the arrow is in the air. When will the arrow hit the ground? th 2256 5400 8256 100 When time = 10 sec, height = 0 feet.

16. Lecture 2816 h = -16t 2 + 160t h = -16t 2 + 160t 100 400 200 300 246 Timeth2256 5400 8256 100 810 Height

17. Lecture 2817 Term: Definitions

18. Lecture 2818 Term: A number or the product of a number and a variable raised to a power. Definitions

19. Lecture 2819 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Definitions

20. Lecture 2820 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Like Terms Definitions

21. Lecture 2821 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Like Terms Terms with the same variables raised to exactly the same powers. Definitions

22. Lecture 2822 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Like Terms Terms with the same variables raised to exactly the same powers. Example: 3x 2, 4x 2 Definitions

23. Lecture 2823 How do we combine like terms?

24. Lecture 2824 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor.

25. Lecture 2825 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor. -8x 2 + 2x – 8 – 6x 2 + 10x + 2

26. Lecture 2826 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor. -8x 2 + 2x – 8 – 6x 2 + 10x + 2

27. Lecture 2827 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor. -8x 2 + 2x – 8 – 6x 2 + 10x + 2 -14x 2 + 12x – 6

28. Lecture 2828 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3)

29. Lecture 2829 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3) 6x 2 + 10x – 20 – 4x 2 + 24x - 12

30. Lecture 2830 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3) 6x 2 + 10x – 20 – 4x 2 + 24x – 12 6x 2 + 10x – 20 – 4x 2 + 24x - 12

31. Lecture 2831 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3) 6x 2 + 10x – 20 – 4x 2 + 24x – 12 2x 2 + 34x – 32 6x 2 + 10x – 20 – 4x 2 + 24x - 12

32. Lecture 2832 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4)

33. Lecture 2833 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4) 10x 2 + 25 – 8x 2 – 2x + 4

34. Lecture 2834 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4) 10x 2 + 25 – 8x 2 – 2x + 4

35. Lecture 2835 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4) 2x 2 – 2x + 29 10x 2 + 25 – 8x 2 – 2x + 4

36. Lecture 2836 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x Cost: C = 3x 2 - 27x + 40

37. Lecture 2837 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x P = (Revenue) – (Cost) Cost: C = 3x 2 - 27x + 40

38. Lecture 2838 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x P = (-5x 2 + 17x) – (Cost) Cost: C = 3x 2 - 27x + 40

39. Lecture 2839 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x P = (-5x 2 + 17x) – (3x 2 - 27x + 40) Cost: C = 3x 2 - 27x + 40

40. Lecture 2840 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x P = (-5x 2 + 17x) – (3x 2 - 27x + 40) Cost: C = 3x 2 - 27x + 40 P = – 5x 2 + 17x – 3x 2 + 27x – 40

41. Lecture 2841 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x P = -8x 2 P = (-5x 2 + 17x) – (3x 2 - 27x + 40) Cost: C = 3x 2 - 27x + 40 P = – 5x 2 + 17x – 3x 2 + 27x – 40

42. Lecture 2842 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x P = - 8x 2 + 44x P = (-5x 2 + 17x) – (3x 2 - 27x + 40) Cost: C = 3x 2 - 27x + 40 P = – 5x 2 + 17x – 3x 2 + 27x – 40

43. Lecture 2843 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x 2 + 17x P = - 8x 2 + 44x - 40 P = (-5x 2 + 17x) – (3x 2 - 27x + 40) Cost: C = 3x 2 - 27x + 40 P = – 5x 2 + 17x – 3x 2 + 27x – 40

44. Lecture 2844 Polynomial: Definitions

45. Lecture 2845 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Definitions

46. Lecture 2846 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 Definitions

47. Lecture 2847 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 3 +x Definitions

48. Lecture 2848 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 3 +x 7+x+5x 6 Definitions

49. Lecture 2849 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 3 +x 7+x+5x 6 x Definitions

50. Lecture 2850 Polynomials:Nonpolynomials: Definitions

51. Lecture 2851 Polynomials:Nonpolynomials: Definitions

52. Lecture 2852 Polynomials:Nonpolynomials: Definitions

53. Lecture 2853 Polynomials:Nonpolynomials: Definitions

54. Lecture 2854 Polynomials:Nonpolynomials: Definitions

55. Lecture 2855 Polynomials:Nonpolynomials: Definitions

56. Lecture 2856 Polynomials:Nonpolynomials: Definitions

57. Lecture 2857 Polynomials:Nonpolynomials: Definitions

58. Lecture 2858 Monomial: Examples: Binomial: Examples: Trinomial: Examples: Definitions

59. Lecture 2859 Monomial: A polynomial with exactly 1 term. Examples: Binomial: Examples: Trinomial: Examples: Definitions

60. Lecture 2860 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: Examples: Trinomial: Examples: Definitions

61. Lecture 2861 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: Trinomial: Examples: Definitions

62. Lecture 2862 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: 2x 3 +9x, 3 +x, 7+5x 6 Trinomial: Examples: Definitions

63. Lecture 2863 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: 2x 3 +9x, 3 +x, 7+5x 6 Trinomial: A polynomial with exactly 3 terms. Examples: Definitions

64. Lecture 2864 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: 2x 3 +9x, 3 +x, 7+5x 6 Trinomial: A polynomial with exactly 3 terms. Examples: 3x 2 +6x+2, 7+5x 3 +4x 2 Definitions

65. Lecture 2865 Degree of a term: Examples:Definitions

66. Lecture 2866 Degree of a term: The sum of the exponents on the variables in the term. Examples:Definitions

67. Lecture 2867 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3

68. Lecture 2868 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3

69. Lecture 2869 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6

70. Lecture 2870 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6

71. Lecture 2871 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x

72. Lecture 2872 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x1

73. Lecture 2873 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x1 xy

74. Lecture 2874 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x1 xy2

75. Lecture 2875 Degree of a polynomial: Examples:Definitions

76. Lecture 2876 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:Definitions

77. Lecture 2877 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9

78. Lecture 2878 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3

79. Lecture 2879 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7

80. Lecture 2880 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6

81. Lecture 2881 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y + 6

82. Lecture 2882 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y + 6 1

83. Lecture 2883 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y + 6 1 xy + 5x - 9y

84. Lecture 2884 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y + 6 1 xy + 5x - 9y 2

85. Lecture 2885 What is a binomial?

86. Lecture 2886 What is a binomial? A polynomial with exactly two terms.

87. Lecture 2887 What is a binomial? A polynomial with exactly two terms. Examples: x + y, 3 + x, 4x 2 + 9

88. Lecture 2888 To Multiply Binomials, use FOIL: (ax+b)(cx+d)

89. Lecture 2889 To Multiply Binomials, use FOIL: acx 2 + (ax+b)(cx+d) F F

90. Lecture 2890 To Multiply Binomials, use FOIL: acx 2 + adx (ax+b)(cx+d) F O O F

91. Lecture 2891 To Multiply Binomials, use FOIL: acx 2 + adx + bcx (ax+b)(cx+d) F O I I O F

92. Lecture 2892 To Multiply Binomials, use FOIL: acx 2 + adx + bcx + bd (ax+b)(cx+d) L F O I L I O F

93. Lecture 2893 To Multiply Binomials, use FOIL: acx 2 + adx + bcx + bd (ax+b)(cx+d) acx 2 + (ad+bc)x + bd L F O I L I O F

94. Lecture 2894 Multiply: (x+5)(x+3) (x+5)(x+3)

95. Lecture 2895 Multiply: x2x2x2x2 (x+5)(x+3) F

96. Lecture 2896 Multiply: x 2 + 3x (x+5)(x+3) F O

97. Lecture 2897 Multiply: x 2 + 3x + 5x (x+5)(x+3) F O I

98. Lecture 2898 Multiply: x 2 + 3x + 5x + 15 (x+5)(x+3) F O I L

99. Lecture 2899 Multiply: x 2 + 3x + 5x + 15 (x+5)(x+3) x 2 + 8x + 15 F O I L

100. Lecture 28100 Multiply: (x-6)(x+2)

101. Lecture 28101 Multiply: (x-6)(x+2) x2x2x2x2 (x-6)(x+2)

102. Lecture 28102 Multiply: (x-6)(x+2) x 2 + 2x (x-6)(x+2)

103. Lecture 28103 Multiply: (x-6)(x+2) x 2 + 2x - 6x (x-6)(x+2)

104. Lecture 28104 Multiply: (x-6)(x+2) x 2 + 2x - 6x - 12 (x-6)(x+2)

105. Lecture 28105 Multiply: (x-6)(x+2) x 2 + 2x - 6x - 12 (x-6)(x+2) x 2 - 4x - 12

106. Lecture 28106 Multiply: (x-7)(x-5)

107. Lecture 28107 Multiply: (x-7)(x-5) x2x2x2x2 (x-7)(x-5)

108. Lecture 28108 Multiply: (x-7)(x-5) x 2 - 5x (x-7)(x-5)

109. Lecture 28109 Multiply: (x-7)(x-5) x 2 - 5x - 7x (x-7)(x-5)

110. Lecture 28110 Multiply: (x-7)(x-5) x 2 - 5x - 7x + 35 (x-7)(x-5)

111. Lecture 28111 Multiply: (x-7)(x-5) x 2 - 5x - 7x + 35 (x-7)(x-5) x 2 - 12x + 35

112. Lecture 28112 Multiply: 6x 2 (2x+5)(3x-8)

113. Lecture 28113 Multiply: 6x 2 - 16x (2x+5)(3x-8)

114. Lecture 28114 Multiply: 6x 2 - 16x + 15x (2x+5)(3x-8)

115. Lecture 28115 Multiply: 6x 2 - 16x + 15x - 40 (2x+5)(3x-8)

116. Lecture 28116 Multiply: 6x 2 - 16x + 15x - 40 (2x+5)(3x-8) 6x 2 - x - 40

117. Lecture 28117 Multiply: 6x 2 - 16x + 15x - 40 (2x+5)(3x-8) 6x 2 - x - 40

118. Lecture 28118 Multiply: (3x+4) 2 9x 2 (3x+4)(3x+4)

119. Lecture 28119 Multiply: (3x+4) 2 9x 2 + 12x (3x+4)(3x+4)

120. Lecture 28120 Multiply: (3x+4) 2 9x 2 + 12x + 12x (3x+4)(3x+4)

121. Lecture 28121 Multiply: (3x+4) 2 9x 2 + 12x + 12x + 16 (3x+4)(3x+4)

122. Lecture 28122 Multiply: (3x+4) 2 9x 2 + 12x + 12x + 16 (3x+4)(3x+4) 9x 2 + 24x + 16

123. Lecture 28123

124. Lecture 28124

125. Lecture 28125