Elements of Algebra

256 ELEMENTS SECT. IV. Here, if we divide by 6, we shall have $x^3 - \frac{11}{6}x^2 + x - \frac{1}{6} = 0$; which equation we may clear of fractions, by the method just explained. First, by making $x = \frac{y}{6}$, we shall have $\frac{y^3}{216} - \frac{11y^2}{216} + \frac{y}{6} - \frac{1}{6} = 0$; and multiplying by 216, the equation will become $y^3 - 11y^2 + 36y - 36 = 0$. But as it would be tedious to make trial of all the divisors of the number 36, and as the last term of the original equation is 1, it is better to suppose, in this equation, $x = \frac{1}{z}$; for we shall then have $\frac{6}{z^3} - \frac{11}{z^2} + \frac{6}{z} - 1 = 0$, which, multiplied by $z^3$, gives $6 - 11z + 6z^2 - z^3 = 0$, and transposing all the terms, $z^3 - 6z^2 + 11z - 6 = 0$; where the roots are $z=1$, $z=2$, $z=3$; whence it follows that in our equation $x=1$, $x=\frac{1}{2}$, $x=\frac{1}{3}$. 725. It has been observed in the preceding articles, that in order to have all the roots in positive numbers, the signs plus and minus must succeed each other alternately; by means of which the equation takes this form, $x^3 - ax^2 + bx - c = 0$, the signs changing as many times as there are positive roots. If all the three roots had been negative, and we had multiplied together the three factors $x+p$, $x+q$, $x+r$, all the terms would have had the sign *plus*, and the form of the equation would have been $x^3 + ax^2 + bx + c = 0$, in which the same signs follow each other *three* times; that is, the number of negative roots. We may conclude, therefore, that as often as the signs change, the equation has positive roots; and that as often as the same signs follow each other, the equation has negative roots. This remark is very important, because it teaches us whether the divisors of the last term are to be taken affirmatively or negatively, when we wish to make the trial which has been mentioned. 726. In order to illustrate what has been said by an example, let us consider the equation $x^3 + x^2 - 34x + 56 = 0$, in which the signs are changed twice, and in which the same sign returns but once. Here we conclude that the equation has two positive roots, and one negative root; and as these