Seminaire Bourbaki 1971-1972, Exposes 400-417 by A. Dold, B. Eckmann

By A. Dold, B. Eckmann

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2) Cancel any factors that are common to a numerator and a denominator: 5 (3) Multiply the remaining factors: 5 5(b 1 3) Answer 4 b 12b2 5(b 1 3) 1 ? (b 1 3) (b 2 3) 3b (b 1 3) 11 1 4b(b 2 3) b. The given fractions and their product are undefined when b 5 0 and when b 5 23. Answer Dividing Rational Expressions We can divide two rational numbers or two rational expressions by changing the division to a related multiplication. Let ba and dc be two rational numbers or rational expressions with b 0, c 0, d 0.

125 14. 6 15. 2 16. 36 17. 108 18. 156 19. 83 20. 57 21. 136 22. pgs 8/12/08 44 1:47 PM Page 44 The Rational Numbers 2-2 SIMPLIFYING RATIONAL EXPRESSIONS A rational number is the quotient of two integers. A rational expression is the quotient of two polynomials. Each of the following fractions is a rational expression: 3 4 ab 7 y 2 2 2 y 2 5y 1 6 a2 2 1 4ab x 1 5 2x Division by 0 is not defined. Therefore, each of these rational expressions has no meaning when the denominator is zero. For instance: 1 5 • x 2x has no meaning when x 5 0.

For example: x 1 5 2x2 5 x 1 5 x 1 1 4x1 1 5 2x2 ? 5 5 (x 1 5)(x 1 1) 10x2 (x 21, 0) This product is in simplest form because the numerator and denominator have no common factors. Recall that a can be written as a1 and therefore, if a 0, the reciprocal of a 1 is a. pgs 8/12/08 1:47 PM Page 51 51 Multiplying and Dividing Rational Expressions EXAMPLE 2 2 2 2 10 5a Divide and simplify: a 2 3a 4a 2 5a 15 Solution How to Proceed (1) Use the reciprocal of the divisor to write the division as a multiplication: a2 2 3a 2 10 5a 2 2 5a 2 10 15 4a 2 5 a 2 3a ?

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