By Rod Downey, Jorg Brendle, Robert Goldblatt, Byunghan Kim
The Asian good judgment convention is the main major good judgment assembly outdoors of North the United States and Europe, and this quantity represents paintings provided at, and coming up from the twelfth assembly. It collects a few fascinating papers from specialists within the box. It covers many components of good judgment.
Readership: Researchers in mathematical common sense and algebra, machine scientists in synthetic intelligence and fuzzy good judgment.
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Additional resources for Proceedings of the 12th Asian Logic Conference
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This is called the standard form of a polynomial. Example 1 Rewriting a Polynomial in Standard Form Polynomial Standard Form Degree a. 4x Ϫ 5x Ϫ 2 ϩ 3x Ϫ5x ϩ 4x ϩ 3x Ϫ 2 3 b. 4 Ϫ 9x 2 Ϫ9x 2 ϩ 4 2 c. 8 8͑8 ϭ 8x0͒ 0 2 3 3 2 ✓CHECKPOINT 1 Rewrite the polynomial 7 Ϫ 9x2 ϩ 3x in standard form and state its degree. ■ Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. 5 Polynomials and Special Products 41 A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0.
32 62. 63. 4 2 Ί 4 ͑3x2͒4 64. Ί 3 x 6 ͑x ϩ 2͒4 66. Ί In Exercises 67–72, simplify the expression. 67. 5Ίx Ϫ 3Ίx 68. 3Ίx ϩ 1 ϩ 10Ίx ϩ 1 69. 5Ί50 ϩ 3Ί8 70. 2Ί27 Ϫ Ί75 71. 2Ί4y Ϫ 2Ί9y 72. 2Ί108 ϩ Ί147 In Exercises 73– 80, use a calculator to approximate the number. ) 3 45 73. Ί 74. Ί57 75. 7 2͞5 76. 1 77. 8 78. 75Ϫ1͞2 3 Ϫ Ί5 2 80. Ϫ4 ϩ Ί12 4 81. Calculator Write the keystrokes you can use to evaluate 4 Ϫ Ί7 in one step on your calculator. 3 82. Calculator Write the keystrokes you can use to evaluate 3 Ί ͑Ϫ5͒5 in one step on your calculator.
Am ϭ amϪn an 3. ͑ab͒m ϭ ambm 4. ab m ϭ am bm 5. ͑am͒n ϭ amn 6. aϪn ϭ 8. ab Ϫn ϭ Product of Powers x7 ϭ x7Ϫ4 ϭ x3 x4 uQotient of Powers ͑5x͒3 ϭ 53x3 ϭ 125x3 Power of a Product 2x Power of a uQotient 3 ϭ 23 8 ϭ 3 x3 x ͑ y3͒Ϫ4 ϭ y3͑Ϫ4͒ ϭ yϪ12 1 an 7. a0 ϭ 1, 32 и 34 ϭ 32ϩ4 ϭ 36 1 y4 yϪ4 ϭ a Definition of negative exponent ͑x2 ϩ 1͒0 ϭ 1 0 ba , Power of a Power 32 n a 0, b 0 Խ Խ ԽԽ Ϫ3 ϭ 23 Definition of zero exponent 3 Խ22Խ ϭ Խ2Խ2 ϭ 22 9. a2 ϭ a 2 ϭ a2 Notice that these properties of exponents apply for all integers m and n, not just positive integers.