Algebra and Trigonometry with Analytic Geometry, Classic by Earl Swokowski, Jeffery A. Cole

By Earl Swokowski, Jeffery A. Cole

The newest version within the hugely revered Swokowski/Cole precalculus sequence keeps the weather that experience made it so well liked by teachers and scholars alike: its exposition is obvious, the time-tested workout units function a number of functions, its uncluttered format is attractive, and the trouble point of difficulties is acceptable and constant. Mathematically sound, ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY, vintage variation, 12E, successfully prepares scholars for extra classes in arithmetic via its first-class, time-tested challenge units

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Extra resources for Algebra and Trigonometry with Analytic Geometry, Classic Edition

Example text

In the next example we illustrate different methods for finding the product of two polynomials. EXAMPLE 3 Multiplying polynomials Find the product: ͑x 2 ϩ 5x Ϫ 4͒͑2x 3 ϩ 3x Ϫ 1͒ SOLUTION Method 1 We begin by using a distributive property, treating the polynomial 2x 3 ϩ 3x Ϫ 1 as a single real number: ͑x 2 ϩ 5x Ϫ 4͒͑2x 3 ϩ 3x Ϫ 1͒ ෇ x 2͑2x 3 ϩ 3x Ϫ 1͒ ϩ 5x͑2x 3 ϩ 3x Ϫ 1͒ Ϫ 4͑2x 3 ϩ 3x Ϫ 1͒ We next use another distributive property three times and simplify the result, obtaining ͑x 2 ϩ 5x Ϫ 4͒͑2x 3 ϩ 3x Ϫ 1͒ ෇ 2x 5 ϩ 3x 3 Ϫ x 2 ϩ 10x 4 ϩ 15x 2 Ϫ 5x Ϫ 8x 3 Ϫ 12x ϩ 4 ෇ 2x 5 ϩ 10x 4 Ϫ 5x 3 ϩ 14x 2 Ϫ 17x ϩ 4.

4 2Ϫ16 is not a real number. 216 Note that 216 Ϯ4, since, by definition, roots of positive real numbers are positive. ” n To complete our terminology, the expression 2 a is a radical, the number a is the radicand, and n is the index of the radical. The symbol 2 is called a radical sign. 3 If 2a ϭ b, then b2 ϭ a; that is, ͑ 2a͒2 ϭ a. If 2 a ϭ b, then b3 ϭ a, or 3 3 ͑ 2 a ͒ ϭ a. Generalizing this pattern gives us property 1 in the next chart. n Properties of 2a (n is a positive integer) Property (1) (2) (3) (4) ͑ 2n a ͒n ϭ a if 2n a is a real number n 2 an ϭ a if a Ն 0 n 2 a ϭ a if a Ͻ 0 and n is odd n n 2 an ϭ ͉ a ͉ if a Ͻ 0 and n is even Illustrations ͑ 25͒2 ϭ 5, ͑ 23 Ϫ8 ͒3 ϭ Ϫ8 2͑Ϫ2͒ ϭ Ϫ2, 2 ͑Ϫ2͒5 ϭ Ϫ2 2͑Ϫ3͒2 ϭ ͉ Ϫ3 ͉ ϭ 3, 2 ͑Ϫ2͒4 ϭ ͉ Ϫ2 ͉ ϭ 2 2 52 ϭ 5, 3 3 3 3 2 2 ϭ2 5 4 If a Ն 0, then property 4 reduces to property 2.

For example, if c Ͼ 0 or if c Ͻ 0 and n is odd, then n 2c nd ϭ 2c n 2 d ϭ c 2d, n n n n provided 2d exists. If c Ͻ 0 and n is even, then n n n 2c nd ϭ 2c n 2d ϭ ͉ c ͉2d, n n provided 2d exists. ILLUS TRATION n Removing nth Powers from 2 2x7 ϭ 2x5 и x2 ϭ 2x5 2x2 ϭ x 2x2 5 5 5 5 5 2 x7 ϭ 2 x 6 и x ϭ 2 ͑x 2͒3x ϭ 2 ͑x 2͒3 2 x ϭ x 2 2 x 3 3 3 3 3 3 2x 2y ϭ 2x 2 2y ϭ ͉ x ͉ 2y 2x 6 ϭ 2͑x 3͒2 ϭ ͉ x 3 ͉ 2 x 6y 3 ϭ 2 x 4 и x 2y 3 ϭ 2 x 4 2 x 2y 3 ϭ ͉ x ͉ 2 x 2y 3 4 4 4 4 4 Note: To avoid considering absolute values, in examples and exercises involving radicals in this chapter, we shall assume that all letters—a, b, c, d, x, y, 22 CHAPTER 1 FUNDAMENTAL CONCEPTS OF ALGEBRA and so on—that appear in radicands represent positive real numbers, unless otherwise specified.

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