By Earl Swokowski, Jeffery A. Cole
The most recent variation 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 purposes, its uncluttered structure is attractive, and the trouble point of difficulties is suitable and constant. Mathematically sound, ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY, vintage version, 12E, successfully prepares scholars for extra classes in arithmetic via its first-class, time-tested challenge units.
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Additional resources for Algebra and Trigonometry with Analytic Geometry, Classic 12th Edition
Using the method of trial and error, as in Example 8, we obtain 4x 2 Ϫ 12xy ϩ 9y 2 ϭ ͑2x Ϫ 3y͒ ͑2x Ϫ 3y͒ ϭ ͑2x Ϫ 3y͒2. Thus, 12x 2 Ϫ 36xy ϩ 27y 2 ϭ 3͑4x 2 Ϫ 12xy ϩ 9y 2͒ ϭ 3͑2x Ϫ 3y͒2. (b) Since each term has x 2y as a factor, we begin by writing 4x 4y Ϫ 11x 3y 2 ϩ 6x 2y 3 ϭ x 2y͑4x 2 Ϫ 11xy ϩ 6y 2͒. By trial and error, we obtain the factorization 4x 4y Ϫ 11x 3y 2 ϩ 6x 2y 3 ϭ x 2y͑4x Ϫ 3y͒͑x Ϫ 2y͒. L If a sum contains four or more terms, it may be possible to group the terms in a suitable manner and then find a factorization by using distributive properties.
Thus, if the set T consists of the first five positive integers, we may write T ϭ ͕1, 2, 3, 4, 5͖. When we describe sets in this way, the order used in listing the elements is irrelevant, so we could also write T ϭ ͕1, 3, 2, 4, 5͖, T ϭ ͕4, 3, 2, 5, 1͖, and so on. If we begin with any collection of variables and real numbers, then an algebraic expression is the result obtained by applying additions, subtractions, multiplications, divisions, powers, or the taking of roots to this collection. If specific numbers are substituted for the variables in an algebraic expression, the resulting number is called the value of the expression for these numbers.
Laws of Radicals Law Illustrations n n n (1) 2 ab ϭ 2 a 2 b ͱ n (2) 250 ϭ 225 и 2 ϭ 225 22 ϭ 5 22 2 Ϫ108 ϭ 2 ͑Ϫ27͒͑4͒ ϭ 2 Ϫ27 2 4 ϭ Ϫ3 2 4 3 ͱ n a 2a ϭ n b 2b 3 mn n (3) ͙ ͙ a ͙a m 3 3 3 3 3 3 5 25 25 ϭ 3 ϭ 8 2 28 3 64 ϭ ͙2 2͑3͒ 264 ϭ 2 26 ϭ 2 6 The radicands in laws 1 and 2 involve products and quotients. Care must be taken if sums or differences occur in the radicand. The following chart contains two particular warnings concerning commonly made mistakes. Y Warning! Y If a 0 and b 0 Illustration (1) 2a2 ϩ b2 a ϩ b 2a ϩ 2b (2) 2a ϩ b 232 ϩ 42 ϭ 225 ϭ 5 3ϩ4ϭ7 24 ϩ 29 ϭ 5 24 ϩ 9 ϭ 213 If c is a real number and c n occurs as a factor in a radical of index n, then we can remove c from the radicand if the sign of c is taken into account.