A treatise on the theory and solution of algebraical equations . of their presence 13G CHAPTER XI. CUBIC EQUATIONS. 202. In the equation x^ — qx — r = 0, the greatest root Xi lies be-tween v/fg and \/4i* 148 204. Formulae for the remaining roots in terms of X\ 149 CHAPTER XII. SYMMETRICAL FUNCTIONS OF THE ROOTS. 207. To obtain the sum of the ma powers of the roots in terms of the coefficients and inferior powers 15G 211. Any rational symmetrical function of the roots of an equationcan be expressed in terms of the coefficients and functions oflower order 150 214. To obtain the equation of the s
A treatise on the theory and solution of algebraical equations . of their presence 13G CHAPTER XI. CUBIC EQUATIONS. 202. In the equation x^ — qx — r = 0, the greatest root Xi lies be-tween v/fg and \/4i* 148 204. Formulae for the remaining roots in terms of X\ 149 CHAPTER XII. SYMMETRICAL FUNCTIONS OF THE ROOTS. 207. To obtain the sum of the ma powers of the roots in terms of the coefficients and inferior powers 15G 211. Any rational symmetrical function of the roots of an equationcan be expressed in terms of the coefficients and functions oflower order 150 214. To obtain the equation of the squares of the differences of the roots of a proposed equation 160 21G-219. Application of symmetrical fmictions to the determination of the roots 102 220, 221. Determination of the values of imaginary roots 164 CHAPTER XIII. ELIMINATION. 227. Elimination by means of symmetrical functions 169 229. On the degree of the final equation 170 231. Elimination by the process for the greatest common measure. 171 232. Improved method of elimination 171 ANSWERS 177. THEOEY AND SOLUTION OF ALGEBRAICAL EQUATIONS INTRODUCTION. 1. Algebraical equations of the first and second degrees aregenerally fully treated of in elementary works on algebra;the student may therefore be supposed to have a knowledgeof the subject so far, and to be acquainted with the meaningof the terms, equation, root, member, &c. The present treatisewill be devoted to a discussion of the general theory of alge-braical equations, and the methods of solution for equationsof the third and higher degrees, sometimes called the HigherEquations. We shall generally express an equation of the nth degreeunder the form, Cnxn + CUz-1 + C2x2 + Cyx + C0 = 0, in which Cr, the coefficient of the rth power of x, is a knownquantity, positive, negative, or zero. The final term, C0, whichmay be regarded as the coefficient of .t°, is frequently calledthe absolute or independent term. The equation is said to be complete when it contains al
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