Elements of Algebra

CHAP. XIII. OF ALGEBRA. 275 cording to the Rule given above, two of these roots are positive, and the two others are negative. 760. But as no root could be determined by this method, when the roots are all irrational, it was necessary to devise other expedients for expressing the roots whenever this case occurs; and two different methods have been discovered for finding such roots, whatever be the nature of the equation of the fourth degree. But before we explain those general methods, it will be proper to give the solution of some particular cases, which may frequently be applied with great advantage. 761. When the equation is such, that the coefficients of the terms succeed in the same manner, both in the direct and in the inverse order of the terms, as happens in the following equation *; $x^4 + mx^3 + nx^2 + mx + 1 = 0$; or in this other equation, which is more general: $x^4 + max^3 + na^2x^2 + ma^3x + a^4 = 0$; we may always consider such a formula as the product of two factors, which are of the second degree, and are easily resolved. In fact, if we represent this last equation by the product $(x^2 + pax + a^2) \times (x^2 + qax + a^2) = 0$, in which it is required to determine $p$ and $q$ in such a manner, that the above equation may be obtained, we shall find, by performing the multiplication, $x^4 + (p+q)ax^3 + (pq+2)a^2x^2 + (p+q)a^3x + a^4 = 0$; and, in order that this equation may be the same as the former, we must have, 1. $p + q = m$, 2. $pq + 2 = n$, and, consequently, $pq = n - 2$. * These equations may be called reciprocal, for they are not at all changed by substituting $1/x$ for $x$. From this property it follows, that if $a$, for instance, be one of the roots, $1/a$ will be one likewise; for which reason such equations may be reduced to others of a dimension one half less. De Moivre has given, in his Miscellanea Analytica, page 71, general formulæ for the reduction of such equations, whatever be their dimension. F. T. See also Wood's Algebra, the Complément des Elemens d'Algebra, by Lacroix, and Waring's Medit. Algeb. chap. 3. T 2