By A. I. Kostrikin, I. R. Shafarevich
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Additional info for Algebra II - Noncommunicative Rings, Identities
Product Formulas Formula Illustration (1) ͑x ϩ y͒͑x Ϫ y͒ ϭ x 2 Ϫ y 2 (2) ͑x Ϯ y͒2 ϭ x 2 Ϯ 2xy ϩ y 2 ͑2a ϩ 3͒͑2a Ϫ 3͒ ϭ ͑2a͒2 Ϫ 32 ϭ 4a2 Ϫ 9 ͑2a Ϫ 3͒2 ϭ ͑2a͒2 Ϫ 2͑2a͒͑3͒ ϩ ͑3͒2 ϭ 4a2 Ϫ 12a ϩ 9 (3) ͑x Ϯ y͒3 ϭ x 3 Ϯ 3x 2y ϩ 3xy 2 Ϯ y 3 ͑2a ϩ 3͒3 ϭ ͑2a͒3 ϩ 3͑2a͒2͑3͒ ϩ 3͑2a͒͑3͒2 ϩ ͑3͒3 ϭ 8a3 ϩ 36a2 ϩ 54a ϩ 27 Several other illustrations of the product formulas are given in the next example. 3 Algebraic Expressions 33 SOLUTION (a) We use product formula 1, with x ϭ 2r 2 and y ϭ 2s: ͑2r 2 Ϫ 2s͒͑2r 2 ϩ 2s͒ ϭ ͑2r 2͒2 Ϫ ͑ 2s͒2 ϭ 4r 4 Ϫ s (b) We use product formula 2, with x ϭ 2c and y ϭ ͩ 2c ϩ 1 ͪ 2 2c ϭ ͑ 2c ͒2 ϩ 2 и 2c и ϭcϩ2ϩ 1 2c 1 2c ϩ : ͩ ͪ 1 2 2c 1 c Note that the last expression is not a polynomial.
For example, (؊3)2/5 can be evaluated successfully as [(؊3)1/5]2 or [(؊3)2]1/5, whereas an error message might otherwise appear. Approximate the realnumber expression to four decimal places. 08͒7/3 3/7 Exer. 93–94: Approximate the real-number expression to four decimal places. 9͒Ϫ2 (b) 527 95 Savings account One of the oldest banks in the United States is the Bank of America, founded in 1812. 04͒180 dollars. Approximate this amount to the nearest cent. 2 2h. Approximate the distance that can be seen from the top of the Chicago Sears Tower, which is 1454 feet tall.
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Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich