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 characteristic numerous functions, its uncluttered structure is beautiful, 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 very good, time-tested challenge units.
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Additional resources for Algebra and Trigonometry with Analytic Geometry, Classic 12th Edition
Are often used to denote sets, and lowercase letters a, b, x, y, . . usually represent elements of sets. Throughout this book, ޒdenotes the set of real numbers and ޚdenotes the set of integers. Two sets S and T are equal, denoted by S ϭ T , if S and T contain exactly the same elements. We write S T if S and T are not equal. Additional notation and terminology are listed in the following chart. Notation or terminology aʦS a S S is a subset of T Constant Variable Meaning Illustrations a is an element of S a is not an element of S 3ʦޚ Every element of S is an element of T A letter or symbol that represents a specific element of a set A letter or symbol that represents any element of a set ޚis a subset of ޒ 3 5 ޚ 5, Ϫ 22, Let x denote any real number 28 CHAPTER 1 FUNDAMENTAL CONCEPTS OF ALGEBRA ͕x ͉ x Ͼ 3͖ is an equivalent notation.
Factoring Formulas Formula (1) Difference of two squares: x 2 Ϫ y 2 ϭ ͑x ϩ y͒͑x Ϫ y͒ (2) Difference of two cubes: x 3 Ϫ y 3 ϭ ͑x Ϫ y͒͑x 2 ϩ xy ϩ y 2͒ (3) Sum of two cubes: x 3 ϩ y 3 ϭ ͑x ϩ y͒͑x 2 Ϫ xy ϩ y 2͒ Illustration 9a2 Ϫ 16 ϭ ͑3a͒2 Ϫ ͑4͒2 ϭ ͑3a ϩ 4͒͑3a Ϫ 4͒ 8a3 Ϫ 27 ϭ ͑2a͒3 Ϫ ͑3͒3 ϭ ͑2a Ϫ 3͓͒͑2a͒2 ϩ ͑2a͒͑3͒ ϩ ͑3͒2͔ ϭ ͑2a Ϫ 3͒͑4a2 ϩ 6a ϩ 9͒ 125a3 ϩ 1 ϭ ͑5a͒3 ϩ ͑1͒3 ϭ ͑5a ϩ 1͓͒͑5a͒2 Ϫ ͑5a͒͑1͒ ϩ ͑1͒2͔ ϭ ͑5a ϩ 1͒͑25a2 Ϫ 5a ϩ 1͒ Several other illustrations of the use of factoring formulas are given in the next two examples.
Thus, (2) is actually two formulas: ͑x ϩ y͒2 ϭ x 2 ϩ 2xy ϩ y 2 and ͑x Ϫ y͒2 ϭ x 2 Ϫ 2xy ϩ y 2 Similarly, (3) represents two formulas. 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.