Elements of Algebra

274 ELEMENTS SECT. IV. an equation, which, multiplied by $6^4*$, becomes $y^4 - 3y^3 + 12y^2 - 162y + 72 = 0$. If we now wish to know whether this equation has rational roots, we must write, instead of $y$, the divisors of $72$ successively, in order to see in what cases the formula would really be reduced to $0$. 758. But as the roots may as well be positive as negative, we must make two trials with each divisor; one, supposing that divisor positive, the other, considering it as negative. However, the following Rule will frequently enable us to dispense with this *. Whenever the signs $+$ and $-$ succeed each other regularly, the equation has as many positive roots as there are changes in the signs; and as many times as the same sign recurs without the other intervening, so many negative roots belong to the equation. Now, our example contains four changes of the signs, and no succession; so that all the roots are positive: and we have no need to take any of the divisors of the last term negatively. 759. Let there be given the equation $x^4 + 2x^3 - 7x^2 - 8x + 12 = 0$. We see here two changes of signs, and also two successions; whence we conclude, with certainty, that this equation contains two positive, and as many negative roots, which must all be divisors of the number $12$. Now, its divisors being $1, 2, 3, 4, 6, 12$, let us first try $x = +1$, which actually produces $0$; therefore one of the roots is $x = 1$. If we next make $x = -1$, we find $+1 - 2 - 7 + 8 + 12 = 21 - 9 = 12$: so that $x = -1$ is not one of the roots of the equation. Let us now make $x = 2$, and we again find the quantity $= 0$; consequently, another of the roots is $x = 2$; but $x = -2$, on the contrary, is found not to be a root. If we suppose $x = 3$, we have $81 + 54 - 63 - 24 + 12 = 60$, so that the supposition does not answer; but $x = -3$, giving $81 - 54 - 63 + 24 + 12 = 0$, this is evidently one of the roots sought. Lastly, when we try $x = -4$, we likewise see the equation reduced to nothing; so that all the four roots are rational, and have the following values: $x = 1$, $x = 2$, $x = -3$, and $x = -4$; and, ac- * This Rule is general for equations of all dimensions, provided there are no imaginary roots. The French ascribe it to Descartes, the English to Harriot; but the general demonstration of it was first given by M. l'Abbé de Gua. See the Memoires de l'Academie des Sciences de Paris, for 1741. F. T.