By A. I. Kostrikin, I. R. Shafarevich

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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.

Express d in miles. ) 58 Fish population The population dynamics of many fish are characterized by extremely high fertility rates among adults and very low survival rates among the young. 00035% of the offspring survive to the age of 3 years. Use scientific form to approximate the number of offspring that live to age 3. 59 Frames in a movie film One of the longest movies ever made is a 1970 British film that runs for 48 hours. Assuming that the film speed is 24 frames per second, approximate the total number of frames in this film.

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Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich

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