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Algebra Factoring Formulas Real numbers : a, b, c Natural number : n 1. a2 – b2 = ( a + b)(a – b ) 2. a3 – b3 = ( a – b )(a2 + ab + b2) 3. a3 + b3 = ( a + b )(a2 - ab + b2) 4. a4 – b4 = (a2 – b2)( a2 + b2) = ( a + b)(a – b ) )( a2 + b2) 5. a5 – b5 = ( a – b)(a4 + a3b + a2b2 + ab3 + b4) 6. a5 + b5 = ( a + b)(a4 - a3b + a2b2 - ab3 + b4) 7. If n is odd, then an + bn = ( a + b)(an-1 – an-2b + an-3b2 - … - abn-2 + bn-1). 8. If n is even, then an – bn = ( a - b)(an-1 + an-2b + an-3b2 + … + abn-2 + bn-1). an + bn = ( a + b)(an-1 – an-2b + an-3b2 - … + abn-2 - bn-1). Product Formulas Real numbers : a, b, c Whole numbers : n, k 9. ( a – b )2 = a2 – 2ab + b2 10. ( a + b )2 = a2 + 2ab + b2 11. ( a – b )3 = a3 – 3a2b + 3ab2 – b3 12. ( a + b )3 = a3 + 3a2b + 3ab2 + b3 13. ( a – b )4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4 14. ( a + b )4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 15. Binomial Formula ( a + b )n = nC0an + nC1an-1b + nC2an-2b2 + … + nCn-1abn-1 + nCn-1bn, 16. ( a + b + c )2 = a2 + b2 + c2 + 2ab + 2ac + 2bc 17. ( a + b + c + … + u + v )2 = a2 + b2 + c2 + … + u2 + v2 + + 2 ( ab + ac + … + au + av + bc + … + bu + bv + … + uv ) Powers Bases (positive real numbers) : a, b Powers (rational numbers) : n, m 18. aman = am + n 19. = am – n 20. ( ab )m = ambm 21. ( 22. (am)n = amn 23. a0 = 1, a 24. a1 = 1 25. a-m = )m = 0 26. = Roots Bases : a, b Powers (rational numbers) : n, m a, b 0 for even roots ( n = 2k, k 27. = 28. = 29. = 30. = ,b = 31. ( )p = 32. ( )n = a 33. = 34. = 35. = 36. 37. ( 0 ,b )m = = ,a 0. 0 N) 38. = 39. = Logarithms Positive real numbers : x, y, a, c, k Natural number : n 40. Definition of Logarithm if and only if x = y= 41. =0 42. =1 43. =- if a > 1 and + 44. = + 45. = - 46. =n 47. = 48. = 49. = 50. x= = , a > 0, a 1. if a < 1 , c > 0, c 1. 51. Logarithm to Base 10 = log x 52. Natural Logarithm = ln x, = 2.718281828… where e = = 53. 54. ln x = 1n x = 0.434294 1n x logx = 2.302585 log x Equations Real numbers : a, b, c, p, q, u, v Solutions : x1, x2, y1, y2, y3 55. Linear Equation in One Variable ax + b = 0 , x = - 56. Quadratic Equation ax2 + bx + c = 0, x1,2 = 57. Discriminant D = b2 – 4ac 58. Viete’s Formulas If x2 + px + q = 0, then x1 + x2 = -p and x1x2 = q 59. ax2 + bx = 0, x1 = 0, x2 = - 60. ax2 + c = 0, x1,2 = 61. Cubic Equation. Cardano’s Formula. y3 + py + q = 0, y1 = u + v, y2,3 = - ( u + v) ( u +v )i, where u= ,v= Inequalities Variables : x, y, z Real numbers : a, b, c, d, a1, a2, a3 …,an, m, n Determinants : D, Dx, Dy, Dz 62. Inequalities, Interval Notations and Graphs a Inequality x b Interval Notation [a, b] a x b (a, b] a x b [a, b) Graph a x x b (a, b) x b, (- , b] x b, (- , b) b x b a x x a , [a, ) a x x a , (a, ) 63. If a > b, then b < a. 64. If a > b, then a – b > 0 or b – a < 0. 65. If a > b, then a +c > b + c. 66. If a > b, then a – c > b – c. 67. If a > b and c > d, hen a + c > b + d. 68. If a > b and c > d, hen a - d > b - c. 69. If a > b and c > d, then ma > mb. 70. If a > b and m > 0, then 71. If a > b and m > 0, then ma < mb. 72. If a > b and m < 0, then 73. If 0 < a < b and n > 0, then an < bn . 74. If 0 < a < b and n < 0, then an > bn . 75. If 0 < a < b, then 76. < > < . . . , where a > 0, b > 0; an equality is valid only if a = b. 77. a+ 2, where a > 0 ; an equality takes place only at a = 1. , where a1, a2, ... , an > 0. 78. 79. If ax + b > 0 and a > 0, then x > - . 80. If ax + b > 0 and a < 0, then x < - . 81. ax2 + bx + c > 0 a>0 a<0 D>0 x < x1, x > x2 x1 < x < x2 D=0 x x1 < x, x > x1 D<0 - 82. <x< x + 83. If < a, then –a < x < a , where a > 0. 84. If < a, then x < -a and x > a, where a > 0. 85. If x2 < a, then < , where a > 0. 86. If x2 > a, then > , where a > 0. 87. If > 0 , then f(x) . g(x) > 0 and g(x) 0 88. If < 0 , then f(x) . g(x) < 0 and g(x) 0 Compound Interest Formulas Future value : A Initial deposit : C Annual rate of interest : r Number of years invested : t Number of times compounded per year : n 89. General Compound Interest Formula A=C(1+ 90. )nt Simplified Compound Interest Formula If interest is compounded once per year, then the previous formula simplifies to : A = C ( 1 + r )t . 91. Continuous Compound Interest If interest is compounded continually ( n A = Cert . ), then