Commensurable Roots of any Equation. 6. Find all the roots of the equation x5-3 x4 8 x3 + 24 x2-9x+27= 0 which has commensurable and equal roots. Ans. 3, -3, and ± 7. Find all the roots of the equation x6 - 23 x4 - 48 x3 +95 x2+400x+375=0 which has commensurable and also equal roots. — 1. Ans. 3, 5, and -2±√ - 1. 313. Problem. To find the commensurable roots of an equation. Solution. Reduce the equation to the form Ax+Bx-1+ &c.... + L x + M = 0, in which A, B, &c., are all integers, either positive or negative. Substitute for z the value and the equation becomes yn Byn-1 x= n-2 Α' A-1+A1+A2 which, multiplied by A-1, is By+++ &c....++M = 0; yn+Byn-1+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 0 Commensurable Roots of any Equation. is not always the one which leads to the most simple result. But when A has two or more equal factors, it is often the case that the substitution leads to an equation of the desired form, A' being the product of the prime factors of A, and each factor need scarcely ever be repeated more than once. 315. EXAMPLES. 1. Find the commensurable roots of the equation 64 x4 328 x3 + 574 x2 - 393x - 90 = 0. Solution. We have, in this case, A = 64 = 26; hence we may take A' equal to some power of 2; and it is easily seen that the third power will do, so that we may make x = y. Hence the given equation becomes у - 41 у3 + 574 у2 - 3144 у + 5760 = 0. The commensurable roots of which are found, as in art 311, to be y = 4, 6, 15, and 16; so that the roots of the given equation are 2. Find the commensurable roots of the equation 8x3+34 x2 - 79 x + 30 = 0. Ans.,, and -6 Commensurable Roots of any Equation. 3. Find the commensurable roots of the equation 24 x3 26 x2 + 9 x - 1 = 0. Ans.,, and 1. 4. Find the commensurable roots of the equation 3 x3 14 x2 + 21 x 10 = 0. Ans. 1, §, and 2. 5. Find the commensurable roots of the equation 8 x4 38 x2 + 49 x2 - 22 x + 3 = 0. Ans., 1, 1, and 3. 6. Find all the roots of the equation 6 x3 +7 x2+39 x + 63 = 0 which has a commensurable root. 7. Find the commensurable roots of the equation 9x6 + 30 x5 +22 x4 + 10 x3 +17 x2 - 20 x + 4 = 0. Ans. and -2. Value of Continued Fractions. CHAPTER IX. CONTINUED FRACTIONS. 316. A continued fraction is one whose numerator is unity, and its denominator an integer increased by a fraction, whose numerator is likewise unity, and which may be a continued fraction. 317. Problem. To find the value of a continued fraction which is composed of a finite number of fractions. Solution. Let the given fraction be |