An Elementary Treatise on Algebra. [With] Key

An Elementary Treatise on Algebra. [With] Key

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1844 edition. Excerpt: ... coefficients real, has imaginary roots, they occur in pairs of the form -+-V-l and 0-0V-I; the two differing only in the sign of the imaginary part. In proving this, we see, in the first place, h&tfx=0 cannot have a single imaginary root: for, if it could have a+a/--1 as a root, without any other imaginary one, then ( 173.) we should have fx = (x--a)(x--b)....(.r-a-oV-T); and, if the actual multiplication were performed, the product of the part, x--a byjx--a)(x--b) would evidently be real, but that of--a'a/--1 by the same would have all its coefficients imaginary. The entire product, therefore, could not be fx, which, by hypothesis, has only real coefficients. Let us now examine, whether there can be two imaginary roots, a4-a'a/--1 and j8 + jSV--1; and if so, what must be the relations of the quantities a, a, l9, and ft'. If these roots be admissible, the product of the factors, x--a--a', /--1 and x--(S--/s'a/--1, must be real. Now, that product is (x--a)(x-P)-(x-)P'-Sl-(x-p)a'/l--a'P'. Of these four terms, the first and last are real: and therefore the entire product will be real, if the two remaining terms can be made to disappear. These terms may be written under the form, --x(a' + P')V+(afi+a'P)V--l.; and this expression will vanish, if a' + /3'=0, and a/3' + a'/9=0. The first of these conditions gives ft'=--a.' and by substituting this in the second, and resolving, we get,3 = a. Hence fx=0 may have two imaginary roots, the second becoming a--a /--1, and the first remaining a--a'/--1, as it was assumed: and, by similar reasoning, it would appear, that, if there be other imaginary roots, there must be two, or some other even number of them; and that each pair must have the relation that has been established. It may be...show more

Product details

  • Paperback | 94 pages
  • 189 x 246 x 5mm | 181g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236640438
  • 9781236640437