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 quite a few purposes, its uncluttered format is attractive, and the trouble point of difficulties is suitable and constant. Mathematically sound, ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY, vintage variation, 12E, successfully prepares scholars for extra 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
And Only a limited number of choices for a, b, c, and d satisfy these conditions. If none of the choices work, then px 2 ϩ qx ϩ r is irreducible. Trying the various possibilities, as depicted in the next example, is called the method of trial and error. This method is also applicable to trinomials of the form px 2 ϩ qxy ϩ ry 2, in which case the factorization must be of the form ͑ax ϩ by͒͑cx ϩ dy͒. EXAMPLE 8 Factoring a trinomial by trial and error Factor 6x 2 Ϫ 7x Ϫ 3. S O L U T I O N If we write 6x 2 Ϫ 7x Ϫ 3 ϭ ͑ax ϩ b͒͑cx ϩ d͒, then the following relationships must be true: ac ϭ 6, bd ϭ Ϫ3, ad ϩ bc ϭ Ϫ7 and If we assume that a and c are both positive, then all possible values are given in the following table: a 1 6 2 3 c 6 1 3 2 Thus, if 6x 2 Ϫ 7x Ϫ 3 is factorable, then one of the following is true: 6x 2 6x 2 6x 2 6x 2 Ϫ Ϫ Ϫ Ϫ 7x 7x 7x 7x Ϫ Ϫ Ϫ Ϫ 3 3 3 3 ϭ ϭ ϭ ϭ ͑x ϩ b͒͑6x ϩ d͒ ͑6x ϩ b͒͑x ϩ d͒ ͑2x ϩ b͒͑3x ϩ d͒ ͑3x ϩ b͒͑2x ϩ d͒ We next consider all possible values for b and d.
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.
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.