Limits of Real Roots. u Secondly. Sturm's theorem gives = 5 23 - 6 x + 2 U" = 3; tot, titi when = 4 it is -, +-, +; so that the equation has three real roots. The row of signs when x = 0 is +,-,-,+; 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 tt, tit; 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. x = 0:1 * = 0.2 X = 0.3 x = 0.4 x = 0.5 X = 0.6 2 = 0.7 2 - 0.8 X = 0.9 titt, ti 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 24 + 8x2 + 16 x 440 = 0. Ans. 3 and - 4. 3. Find the first approximation to the roots of the equation 25 — 15 23 + 132 x2 + 36 x + 396 = 0. Ans. 1,-1,- 5. 4. Find, by Stern's theorem, the greatest possible number of real roots which the equation 210 10 28 24 + x — 11=0 can have between + 1 and 1. Snlution. In this case we have, by art. 294, 10 28 11 560 26 12 22 = 151200 24 201600 22 Uix = 3628800 ; S-,-,-, tet, titi when x = -1 it is -, +,-, +,-, +,-,-, t-ti so that the number of these roots cannot exceed 8. u = 210 24 x anto 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 it,, -,--, t, -; 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 a = 0, that it is when = 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 -, t-, +-+-+--+; 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 x + 23 — 22-1=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 rn tarn-1 +brn- + &c. +lo+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, P 9 Method of fiuding Integral Roots. As this root must verify the given equation, we have pr-2 + &c. + m = 0; p" p-1 +6 " whence, multiplying by qu-1, and transposing, we obtain p" a p-1-aq- &C.... mqn-1; 9 and, therefore, as the second member is integral, the first member must also be integral, or we must have q=1, whence x = p; that is, every commensurable root of the given equation must be an integer. Again, the substitution of X = P, in the given equation, produces pa + apa-' + &c. ... + kp? +1p+m= 0; whence, dividing by p, and transposing, we obtain m 1 - kp- &c. ... apa-2-pa-1; р and, therefore, as the second member is integral, the first member must be so likewise; that is, every integral root must be a divisor of m. If, now, we denote by m', = + m = р the preceding equation gives, by transposing and dividing by p, m' =-ap -k-ip-hp?—g p3— &c.—apa-3-p*-4. р so that this integral root must likewise be a divisor of m'. Method of finding Integral Roots. In the same way, if we use m', m'", miv, &c. as follows: m' + k, р m" m" = ti, P mm = +h, P &c. ; this integral root must be a divisor of m", m'", mr, and the last condition to be satisfied is min-1)+p=0, or min-1)=-p. Hence to find all the commensurate roots of the given equation, write in the same horizontal line all the integral divisors of m, which are contained between the extreme limits of the roots. Write below these divisors all the corresponding values of m', m", fc., which are integral, remembering that a divisor cannot be a root, when the value which it gives for either m', m“, m'", foc., is fractional. Proceed in this way till the values of min-1) are obtained, and those divisors only are roots which give -p for the value of this quantity. 312. EXAMPLES. 1. Find the commensurable roots of the equation 25 — 19 23 + 34 22 + 12 x 40 = 0. Solution. The extreme limits of the real roots are +704, and 6.9. Hence we have |