Limits of Negatiye Roots. If we, then, denote by L this superior limit of the positive roots, we have that is, a superior limit of the positive roots is unity, increased by that root of the greatest negative coefficient, whose index is equal to the excess of the degree of the equation above the exponent of the first nega tive term. 306. Problem. To find an inferior limit of the positive roots. Solution. Substitute in the given equation for x, the value x= and find, by the preceding article, a superior limit of the positive values of y, after the equation is reduced to the usual form; and denote this limit by L'. is an inferior limit of the positive roots of the given equation. 307. Problem. To find the limits of the negative roots of an equation. Solution. Substitute for x Limits of Real Roots. and the positive roots of the equation thus formed are the negative roots of the given equation; and, therefore, the limits of its positive roots become, by changing their signs, the required limits. 308. Corollary. By the substitution of different. numbers for p and q, in arts. 290 or 294, the limits between which each root is obtained can be narrowed to any extent which may be desired, until they may be adopted as the first approximations to the roots in the method of art. 179. Thus, it is easy to obtain the first left hand significant figure. 309. EXAMPLES. 1. Find the left hand significant figure of the real roots of the equation Solution. 5x36x+2=0. First. In this case, 6 is the greatest negative coefficient and 6 x is the first negative term, so that, by art. 305, 1+6=3.5 is a superior limit of the positive roots. To find the limit of the negative roots, let -y, and the equation becomes, by reversing its signs, so that 5 y3-6 y 2 =0; ყვ - (1+√6) = −3·5 is the superior limit of the negative roots, and the roots are all contained between 4 and — 4. 1 so that the equation has three real roots. The row of signs when x = 0 is +, 一, 一, + ; so that two of the roots are positive and one is negative. The substitution of positive integers, gives for the rows of signs when x = 1 so that both the positive roots are contained between 0 and 1. The substitution of the positive decimals 0.1, 0·2, 0·3, · · &c., gives the following rows of signs. so that one real root is contained between 0-3 and 0.4, and the other between 0.8 and 0.9; their first approximate val ues are, then, 0·3 and 0·8. The substitution of the negative integers gives, in the Limits of Real Roots. same way-1, for an approximate value of the negative root. 2. Find the left hand significant figures of the roots of the equation x2 + 8 x2 + 16 x 440 0. 3. Find the first approximation to the roots of the equa tion 25-15 x3 132x2+36x+396 0. Ans. 1,-1,- 5. 4. Find, by Stern's theorem, the greatest possible number of real roots which the equation so that the number of these roots cannot exceed 8. Integral Root. Again, when x is infinitely little greater than zero, for which value some of the differential coefficients vanish, the row of signs is so that there cannot be more than three roots between 0 and 1; and since the sign of the first term is the same when x = 0, that it is when x = 1, there cannot, by art. 283, be an odd number of real roots between 0 and 1, and consequently there cannot be more than 2. The row of signs when x is less than zero by an infinitesimal is so that there can be no real root between 0 and · 1. 5. Find, by Stern's and Descartes' theorems, the greatest possible number of real roots of the equation 26. - 5 x2 + x3 — x2 — 1 — 0. comprised between 0 and 1. Ans. 2. 310. A Commensurable Root is a real root, which can be exactly expressed by whole numbers or fractions. 311. Problem. To find the commensurable roots of the equation x2+ax"-1+bx"−2+ &c. + 1x+m=0, in which a, b, &c. are all integers, either positive or negative. Solution. Let one of the commensurable roots be, when reduced to its lowest terms, |
