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 apparent, the time-tested workout units characteristic quite a few 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 version, 12E, successfully prepares scholars for extra classes in arithmetic via its very good, time-tested challenge units.

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**Example text**

This technique, called factoring by grouping, is illustrated in the next example. 38 CHAPTER 1 FUNDAMENTAL CONCEPTS OF ALGEBRA EXAMPLE 10 Factoring by grouping Factor: (a) 4ac ϩ 2bc Ϫ 2ad Ϫ bd (b) 3x 3 ϩ 2x 2 Ϫ 12x Ϫ 8 (c) x 2 Ϫ 16y 2 ϩ 10x ϩ 25 SOLUTION (a) We group the first two terms and the last two terms and then proceed as follows: 4ac ϩ 2bc Ϫ 2ad Ϫ bd ϭ ͑4ac ϩ 2bc͒ Ϫ ͑2ad ϩ bd͒ ϭ 2c͑2a ϩ b͒ Ϫ d͑2a ϩ b͒ At this stage we have not factored the given expression because the right-hand side has the form 2ck Ϫ dk with k ϭ 2a ϩ b.

ILLUS TRATION The Notation can 5 и 23 ϭ 5 и 8 ϭ 40 Ϫ5 и 23 ϭ Ϫ5 и 8 ϭ Ϫ40 Ϫ24 ϭ Ϫ͑24͒ ϭ Ϫ16 3͑Ϫ2͒3 ϭ 3͑Ϫ2͒͑Ϫ2͒͑Ϫ2͒ ϭ 3͑Ϫ8͒ ϭ Ϫ24 We next extend the definition of an to nonpositive exponents. 2 Exponents and Radicals 17 If m and n are positive integers, then m factors of a ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ aman ϭ a и a и a и и и и и a и a и a и a и и и и и a. n factors of a Since the total number of factors of a on the right is m ϩ n, this expression is equal to amϩn; that is, aman ϭ amϩn. We can extend this formula to m Յ 0 or n Յ 0 by using the definitions of the zero exponent and negative exponents.

For example, x 2 Ϫ 2 is irreducible over the rational numbers, since it cannot be expressed as a product of two polynomials of positive degree that have rational coefficients. However, x 2 Ϫ 2 is not irreducible over the real numbers, since we can write x 2 Ϫ 2 ϭ ͑ x ϩ 22͒͑ x Ϫ 22͒. 4, not over the complex numbers. Every polynomial ax ϩ b of degree 1 is irreducible. Before we factor a polynomial, we must specify the number system (or set) from which the coefficients of the factors are to be chosen.