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 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, Method of finding Integral Roots. As this root must verify the given equation, we have + &c. +m=0; whence, multiplying by q-1, and transposing, we obtain pr == - a p" — 1 — b p2 −2 q — &c. ... and, therefore, as the second member is integral, the first member must also be integral, or we must have whence q = 1, x = pi that is, every commensurable root of the given equation must be an integer. Again, the substitution of x = P, in the given equation, produces p" + a p"−1 + &c. ...+kp2+1p+m=0; whence, dividing by p, and transposing, we obtain m Р =—l — kp — &c. ... — a p"-2 — pr−1 ; 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. m' -3 - k — i p—h p2 —g p3—&c.—a pr—3—p”—§. 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'"', m1v, &c. as follows: this integral root must be a divisor of m", m", m3, &c. ; and the last condition to be satisfied is m[n-1]+p=0, or m[n—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", &c., which are integral, remembering that a divisor cannot be a root, when the value which it gives for either m', m", m"", &c., 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 25-19x334 x2 + 12x-40 = 0. Solution. The extreme limits of the real roots are +7·4, and -6.9. Hence we have - 1, and 1,-28, 1, 14, 22, 20; 30; -25; 5; and, therefore, 2, -5 are roots of the given equation, and its first member, divided by the factor (x-2)(x+1)(x+5) = x3 + 4x2-7x-10, and, therefore, the remaining roots are those of the equation which are equal to each other, and each is x = 2. 2. Find the commensurable roots of the equation x3-3x2-10 x6—2x2+6x3+21 x2-3x-10=0. Ans. 5, 1, -1, and —2. 3. Find all the roots of the equation x4x3-24 x2 +43x-21=0. 5. Find all the roots of the equation x4—10 x3 +35 x2 — 50 x + 24 = 0 which has commensurable roots. Ans. 1, 2, 3, 4. Commensurable Roots of any Equation. 6. Find all the roots of the equation x53x48x3+24 x29x+27=0 which has commensurable and equal roots. Ans. 3, -3, and ±√−1. 7. Find all the roots of the equation x6-23 x4-48 x3+95 x2+400x375=0 which has commensurable and also equal roots. Ans. 3, 5, and -2-1. 313. Problem. To find the commensurable roots of an equation. Solution. Reduce the equation to the form AxBx-1+ &c.... + Lx+M=0, in which A, B, &c., are all integers, either positive or negative. yn Byn-1 Cy-2 Ly + + + &c.... ++ M=0; An-1 An-1 An-2 which, multiplied by A-1, is y+By+A Cyn-2+&c....+An-2 Ly+An¬1 M = 0. The commensurable roots of this equation may be found, as in the preceding article, and being divided by A, will give the commensurable roots of the required equation. 314. Scholium. The substitution of |