By Raymond Smullyan
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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.