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 apparent, the time-tested workout units function various functions, its uncluttered structure 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 additional classes in arithmetic via its first-class, time-tested challenge units.
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Extra info for Algebra and Trigonometry with Analytic Geometry, Classic 12th Edition
We often arrange the terms of a polynomial in order of decreasing powers of the variable; thus, we write 2 2 5z 2 2 5z Ϫ 3z7 ϩ 8 Ϫ 25 z 4 ϭ Ϫ3z7 Ϫ 25 z4 ϩ 25 z2 ϩ 8. We may regard a polynomial in x as an algebraic expression obtained by employing a finite number of additions, subtractions, and multiplications involving x. If an algebraic expression contains divisions or roots involving a variable x, then it is not a polynomial in x. 1. In particular, if additions, subtractions, and multiplications are carried out with polynomials, we may simplify the results by using properties of real numbers, as demonstrated in the following examples.
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.
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.