By George Chrystal
This Elibron Classics publication is a facsimile reprint of a 1904 version by way of Adam and Charles Black, London.
Read Online or Download Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 PDF
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This Elibron Classics ebook is a facsimile reprint of a 1904 variation by means of Adam and Charles Black, London.
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Additional info for Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1
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.