Bonnycastle's Introduction to Algebra

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1851 edition. Excerpt: ... equation. Here a = 0, 5 = 0, c = 12, and Z =--17; Whence, by substituting these numbers in the cubic equation, z-(Lb2 + d) z =--b3 + i c2-i 5d, v12; 108 8 3 ' we shall have, after simplifying the results, ' + 17= 18, Where it is evident, by inspection, that z = 1. And if this number be substituted for r9 0 for b, and--17 for d in the two quadratic equations in the above rule, their solution will give aJ==_iv2+V(--+ u 18)=--V2 + V(--+3V2) aj=-iV2-V(--i + V18)=-V2--V(--i + 3 V-2) = +iV2+V(--4--. V18) = +V2 + V(----3V2) a;=+iV2--V(----V18J=+iV-2--V(----3V2) Which are the four roots of the proposed equation; the first two being real, and the last two imaginary. Rule 3.--The roots of any biquadratic equation of the form #4-f-act2-j-bx--c = 0, may also be determined by the ratios, a#--px--q = Qy and c&--rx-j-s = 0, or its equal #2--px--s = 0 we shall have z =--hpdh V(jp2--q)'y x = hp: V(p--s)y which expression, when taken in + and--, give the four roots of the proposed biquadratic as was required. This method, which differs'considerably from either of the former, consists in supposing the root of the given equation, xi + ax2-f-bx-j-c = 0 (1), to be of the following trinomial surd form x--Vp--Vq-Vr where p, q, r: denote the roots of the cubic equation of which the coefficients/, g, and the absolute term A, are the unknown quantities that are to be determined. following general formulae first given by Etjler; which are remarkable for their elegance and simplicity. Find the three roots of the cubic equation z5 + 2az2 + a2--4c) z = b2, by one of the former rules before given for this purpose; and let them be denoted by r r," and r'" Then, agreeably to the theory of equations before given, we shall have p--q--r----/ j...