| XXX | (115) | XXX |
A L G E Examp. Suppofe 8vJ—26x-i+ii.v+io=Oj and that it is required to find the values of x} the operation is thus; Divifors.Progr Jlefultf. Suppof. , . C+3 i.3“26/V*-lri i*+'io— A +10 .2,.5,1.0. == o —i' C—3? '5»7»35 The difference of the terms of the laid arithmetical progreffion is 2, a divifor of 8, the coefficient of the higheft term x% of the .equation, therefore fuppofitig m~2, J, we try the divifor 2x—5 ; which fucceeding, it follows, that 2X—5=0, or x—2i. The quotient is the quadratic qx3—3X—2=0, whofe ‘roots are and fo that the three roots of the propofed equation are 2^, 2—, 3—V41 The other arithmetical progrefiion gives x+2 for a divifor ; but it does not fucceed. If the propofed equation has no fimple divifor, then we are t© inquire if it has nor fome quadratic divifor (ifitfelf is an equation of more than three dimenfions.) An equation having the divifor wx2—nx--r may be -exprefled, as in the firft article of this chapter, by Ey.mx1' — n. -f-v ; and if we fubftitute for x any1 known quantity a, the fum that will refult will.have mu —na-~r for one of its divifors; and,' if we fubflitute fucceffively for x the progre (Ron a, a—f, —2?, a—y, &c„ the fums that arife from this fubilitution will have ma'l—ria--r -nY.a — e--r —nY.a—2e+r m'Aa—y z—«X.;—3?+r, among their divifors refpedively. Thefe terms are not now, as in the laft cafe, in arithmetical progreffion ; but if you fubtrad them from the fquares of the terms a, a—e, a—2e, •—y. Sic. multiplied by nt a divifor of the higheft term of the propofed equation, thatz is from ma mbba-r—e * wiYa—:<* 11 vi'Yci—jT) , &c. the remainders. na—r nXa—e—r wXrf—2e—r
- n--a—7,e-—r, See. ffiall be in arithmetical
progreffion, having their common difference equal to ,nXe. If, for example, we fuppofe the affirmed progr.effion a, a—e. a—2e a—y> See. to be 2, x, o, —x, the di-vifors will be t
l A. R 5 4mrn—2«+r — « -1-r . +r ?n + n + r, which fubtrafted from 4;;?, m, o, m, leave 2«—r
B
—n—r, an arithmetical progreflion whofe difference is +«; and whofe term, arifing from the fubftitution of o for x, is —r. From which it follows, that by this operation, if the propofed equation has a quadratic divifor, you will find an arithmetical progreifion that will determine to you n and r, the coefficient ?n being fuppofed known ; fince it is unit, or a divifor of the coefficient of the higheft term of the. equation. Only you are to obferve, that if the firft term w/xz of tRe quadratic divifor is negative, then in order to obtain an arithmetical progreffion, you are not to fubtrad, but add the divifors —Am—2nr, —m—+r, —to the terms 4772, m, o, m. The general rule therefore, deduced from what we had faid, is, “ SuBftitute in the propofed equation for x the terms “ 2, 1,0, —1, fucceffively. ’ Find all the divifors “ of the fums that refult, adding and fubtrading them “ from the fquares of thefe numbers 2, 1,0, —1, foe. “ multiplied by a numerical divifor of the higheft term “ of the propofed equation, and take but all the arith“ metical progreffions that can be found amofigft thefe “ fums and differences. Let r be that term in any' “ progrellion that arifes from the fubftitution of x—o, “ and let =+=« be the difference arifing from fubtrading “ that term from the preceding terrain the progreffion ; “ laftly, let m be the forefard divifor of the higheft. “ term; then ffiall mx*z±znx—r be the divifor that “ ought to be tried.” And one one or other of the divifors found in this manner will fucceed, jf the propofeef equation has a quadratic divifor. Chap. XXI. Of the Method* by which yen may approximate to the Roots of Numerical Equations by their Limits. When any equation is propofed to be refolved, firffi find the limits of tire roots (by chap, vf.) as for example, if the roots of the equation x2—i6x--^=<y are required, you find the limits are b, 8. and 17, by p. no. col. 2. par. 2.: that is, the leaft root is between o and 8, and the greateft between 8 and 17. In order to find the firft of the roots, I confider, that if I fubftitute o for x in > 2—idx+yy,; the refult is pefitive, viz. +55, and confequently any number betwixt c and 8 that gives a- pofitive refult, mu.ft be lefs than the leaft root, and any number- that gives a negative refult mult be greater. Since o and ,8 are the limits, I try 4,x that 2is, the mean betwixt them, and fuppofing — 4> * that —64+55=7, whichnowL conclude the root is greater than 4. fromSo that we have the root limited between 4 and. 8. Therefore I
