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 rig A L G E B R A. Where the labour is very fhort, fince We need only atSimple divifors, 2, 3j 5* • tend to the firft expreffion; and we fee immediately that The produfts of two,. . 6, 9, 10., .15. 4 fubftituted for x gives a pofitive r.efult, whence all the The produets of three,. , 18, 30, 45. divifors of 72 that exceed 4 are to be rejected;' and thus The produdt of all four, 90. by a few trials, we find, that 4*2 is the pofitive root of the equation'. Then dividing the equation by *—2, and The divifors of nabb. refolving the quadratic equation that is the quotient of The fimple divifors, 3, 7, b, b. The produtts of two, 21, 3#, 3/^, 7^, 7$, ab, bb; the diviiion, you find the other two roots to be —9, and The produils of three, • sitf, 21^' '^ab, ^hb; 'jab, •jbh, abb. But there is another method that reduces the divifors of the laft term, that can be ufeful,. ftill to more narrow The produdts of four, 2iab, 2lbb, .^abb, Jabb. The produdts of the five, ... . 2iabb. limits, Suppofe the cubic equation x^—px'+qx—r—o is pfoBut as the laft term may have very many divifors, and pofed to be refolved. Transform it to an equation whole the labour maybe very great to fublKtute them all for roots ffiall be lefs than the values of * by. unity, aifuthe unknown quantity, we (hall now (hew how it may be ming' >—*—x. And the laft term of the transformed abridged, by limiting to afmall number the divifors you equation will be i—p+q—r; .which is found by fubftiare to try. And, firft, it is plain, from p. 109. col. 1. tuting unit, the difference of x and9, for x, in the propur. 4. that “ any divifor that exceeds the greateft nega- pofed equation; as will eafily appear from p. 106. col. 1 the laft term of the tranff‘ tive coefficient by unity is to be neglcdted.” Thus, in par. 4. where, when 9=*—e, 3 refolving the equation ur4—2*3—2 5vI+26x+i20=o, as forrhed equation was e -—pe^-^cqi—r. t 25 is the'greateft negative coefficient, we conclude, that Transform again the equation —p --q- —r-—0, by the divifors of 120 that exceed 26 may.be negleded. affuming y—A-f-i, into an equation whofe roots ffiall exBut the labour may be ftill abridged, if we make ufe of ceed the values of * by unit, and the laft term of the the rule in the beginning of ch. i£.; that is, if we find the transformed equation will be —1—p-—q—r, the fame number which fubilituted in thefe following exprellions, that arifes by fubftituting — t, the difference betwixt x and^, for x, in the propofed equation. x*—2-r3—2 5.v14-26x+i20, Now the values of x are feme of the divifors of r, 2*31—3 V*—25V +13, which is’the term left when you fuppofe x=o ;• and the 6* ——25, values of the /s are feme of the divdors of +1—p+q 2x —I, —/•, and of —1—/>—y—r, refpeedively. And thefe values are in arithmetical progreffion increafing by the will give in them 1 all a pofitive refult: for that number common difference unit; becaufe x—1, x, x-ri, are in will be greater than the greateft root, and all. the divifbrs that progreflion. And it is obvious the fame reafonirig of 120 that exceed it may be negleded. be extended to any .equation of whatever degree. That this inveftigation may be eafier, wejought to be- may So that this gives a general method for the refoiution of gin always v/ith that expreffion where the negative roots equations whole roots are commenfurabie. feem to prevail moft; as here in the quadratic expreflion Subftitute, in place of the unknown quantity, —6x—25; where finding that 6 fubftituted for x Rule. gives that expreffion pofitive, and gives all the other ex- fucceffively the terms of the prpjyeffion, 1, o, —1, preffions at the fame time pofitire, I conclude, that 6 is &c. and find all the divifors of the fums that refult; greater than any of the Toots, and that all the divifors then take out all the arithmetical progreffions you can find among thefe divifors, whofe common difference is of 120 that exceed 63 may be negle&ed. If the equation ^r oat—72=0 is propofed, unit; and the values of x will be among the divifors arifing from the fubftitutions of x=o that belong to the rule of p. 109. col. 1. par. 4. does not help to abridge thefe The values of x will be affirmative the operation; the laft term itfelf being the greateft nega- when progre0ions. the arithmetical progreffion increafes, but negative term. But, by chap. 18.. we inquire'what number tive when it decreafes. fubftituted for x will give all thefe expreffions pofitive. Examp. "Let it be required to find one of the roots x'+ux^+iox—72of the equation x3—x*—iox-)-6=o. The operation is 2,X1-+22x--lQ' thus; 3*-f-li.

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3, 2 ; the term of which, oppofite to the fuppofition Where the3 fuppofitions of x=i, x=6> x=—1, give the of4, x=G, being 3, and the feries decreafing, we try if quantity x —-x*—iox4-6 equal to—4, 6, 14; among •—3 fubftituted for x makes the equation vanilh; which v/hofe divifors we find only one arithmetical progreflion Ff , fucceeding Vol. I. No. 5. 3