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### 32 Cards in this Set

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 Factor x²-y² (x + y)(x - y) Factor x²+y² No real factorization Using complex numbers, with i = sqrt(-1) (x+yi)(x-yi) Factor x³+y³ (x+y)(x²-xy+y²) Factor x³-y³ (x-y)(x²+xy+y²) Factor x²+2xy+y² (x+y)² Pythagorean Theorem For a right triangle with legs of length a and b, and hypotenuse of length c a²+b²=c² Perfect Squares [0,50] 0²=0 1²=1 2²=4 3²=9 4²=16 5²=25 6²=36 7²=49 Perfect Squares [50,250] 8²=64 9²=81 10²=100 11²=121 12²=144 13²=169 14²=196 15²=225 Powers of 2 [1,500] 2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 Perfect Cubes [1,1000] 1³=1 2³=8 3³=27 4³=64 5³=125 6³=216 7³=343 8³=512 9³=729 Factorials [1,360000] 0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 Factor x^4 - y^4 (x-y)(x+y)(x²+y²) Factor x^5 - y^5 (x - y)(x^4 + x³y + x²y² + xy³ + y^4) Factor x^5 + y^5 (x + y)(x^4 - x³y + x²y² - xy³ + y^4) Factor x^6 - y^6 (x-y)(x+y)(x²+xy+y²)(x²-xy+y²) Factor x^6 + y^6 (x² + y²) (x^4 - x²y² + y^4) ax² + bx + c = 0 i) Discriminant ii) Quadratic Formula i) D=b²-4ac ii) x = [ -b ± sqrt(D)] / 2a D < 0 -> complex roots ax²+bx+c=0 Properties of the Discriminant D = b² - 4ac i) D>0 and a perfect square -> 2 rational roots ii) D>0 and not a perfect square -> 2 irrational roots iii) D=0 -> 1 root, multiplicty 2 iv) D<0 -> 2 complex conj. roots Primative Pythagorean Triples with perimeter < 100, a b c p 3 4 5 12 5 12 13 30 8 15 17 40 7 24 25 56 20 21 29 70 12 35 37 84 9 40 41 90 Heron's Formula for the area A of an arbitrary triangle with side lengths a, b, c, and semiperimeter s A² = s(s-a)(s-b)(s-c) -or- A = sqrt[s(s-a)(s-b)(s-c)] Primes [2,197] 2 29 67 107 157 3 31 71 109 163 5 37 73 113 167 7 41 79 127 173 11 43 83 131 179 13 47 89 137 181 17 53 97 139 191 19 59 101 149 193 23 61 103 151 197 Digits of pi 3.14159 26535 89793 23846 26433 83279 50288 4197 ... irrational and transcendental unknown whether normal Digits of e 2.71828 18284 59045 23536 02874 71352 66249 7757... Known to be both irrational and transcendental; e is maximally transcendental, with irrationanality measure, mu(e)=2. it is unknown whether e is normal The Golden Ratio phi 1.61803 39887 49894 84820 45868 34365 63811 7720... irrational and algebraic phi = ½ [ 1 + sqrt(5) ] The Euler-Mascheroni constant gamma 0.57721 56649 01532 86060 65120 90082 40243 1042... unknown if irrational Catalan's Constant K K = 0.915965594177... unknown whether irrational Apéry's constant zeta(3) zeta(3) = 1.2020569... known to be irrational not known to be transcendental or normal Pascal's Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Perfect Number a # equal to the sum of its factors 6 28 496 8128 33550336 8589869056 137438691328 2305843008139952128 The Hardy-Ramanujan Number The smallest "taxicab number", a number that has two representations as the sum of two cubes. 1729 = 1³+12³ = 9³+10³ Platonic Solids Solid f v e Tetrahedron 4 4 6 Cube 6 8 12 Octahedron 8 6 12 Dodecahedron 12 20 30 Icosahedron 20 12 30 Note f+v-e=2 The Kepler-Poinsot Solids great dodecahedron great icosahedron great stellated dodecahedron small stellated dodecahedron