...all degrees to be produced by a multiplication of binomial factors. Every affefted equation will have as many roots as there are units in the exponent of the higheft power of the variable quantity ; and, if the terms of the equation are alternately affirmative...

...that which characterizes these roots is, that on substituting each of them successively instead of x+ the aggregate of the terms of the equation vanishes...exponent of the highest power of the unknown quantity, und that each root has the properly of rendering, by its substitution in place of the unknown quantity,...

...degree may be considered as proekiced by the multiplication of аз many simple equations ns there arc units in the exponent of the highest power of the unknown quantity. From this he deduced the relation which exists bctw en the roots of an equation, and the coefficients...

...four roots or values of x ; and that which characterizes these roots is, that on substituting eacb of them successively instead of r, the aggregate of...of the highest power of the unknown quantity, and thnt each root h;is the property of rendering, by its substitution in place of the unknown quantity,...

...aggregate of the terms of the equation vanishes, by the opposition of the signs + and — . ' 1 he preceding equation is only of the fourth power or...quantity, the aggregate of all the terms of the equation equul to nothing. It must be observed that we cannot have all at once x = a, x = b, x = c, &c. for...

...is only of the fourth power or degree ; but it is manifest that the above remark applies i<equations of higher or lower dimensions : viz. that in general...rendering, by its substitution in place of the unknown quantify, the aggregate of all Ihe terms of the equation equal to nothing. It must be observed that...

...law should be remembered. Second Property. 264. Every equation involving but one unknown quantity, has as many roots as there are units in the exponent of its degree, and no more. Let the proposed equation be if+Par-i+Q«" 3+ • • . +Tx+\J=0. Since every...

...law should be remembered. Second Property. 281. Every equation involving but one unknown quantity, has as many roots as there are units in the exponent of its degree, and no more. Let the proposed equation be xn+Pxm~l+Q.xm-2+ . . . +Ta;+U=0. Since every...

...all arranged on one side, the polynomial we thus get is composed of as many simple binomial factors as there are units in the exponent of the highest power of the unknown quantity. The discovery of these factors would he the discovery of the roots of the equation, since these are...

...on of the divisors of all degrees. 234. As an exemplification of the principle, that every equation has as many roots as there are units in the exponent of the highest power of the unknown quantity, we propose to examine the equation xm—! =0. Let us commence by making m=2, and we have x2 — 1=0By...