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 114 ALGEBRA. fucceeding one of its roots muft be —3. Then divi- 3 If it is required to find the roots of the equation ding the'equation by1 we find the roots of the x — 3** — — ']2 = b, the operation will be (quadratic) quotient x —4X-|-2=o are 2. thus ; SuppofR'futs P rogreffions. X = I; 120 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120 8 3 4 .5 = 0— 72 i,2,^3,446,8,9,12,18,24j36,72. 9234 =—t— 301,2,3,5,6,10,15,30. ■O I 2 3 7na—71, Of thefe four arithmetical progreflions having their via—-me-—71, common difference equal to unit, the firft gives x=9, the ma—277ie—n, which are alfo'in arithmetical proothers give x——2, x=-^3, x==—4; all which fucceed except x=—3 : fo that the three values of x are greffion, having their common difference equal to me. If, for example, we fubflitute for x the terms'of this +9> —2, —4. progreifion, 1, o, —1, the quantities that refult have among their divifors the arithmetical progreffion rn—n, —m—n ; or, changing the figns, 7,—m, n, n--7n ‘ Chap. XX. 0/ the Refolution of Equations —n, the difference of the terms is m, and the term by/finding the Equatfons of a lower Degree Where belonging to the fuppofition of x=o is n. that are their Divifors. It is manifefl therefore, that when an equation has any if you^fubflitute foryv the prdgreffion 1, To find the roots of an equation is the fame thing as fimple divifbr, will be ?ound amongfl the divifors of the to find the fimple "equations, by the multiplication of o,fums—1,thatthere refult from thefe fubltfuitions, one arithmetical which into one another it is produced, or, to find the progreffion at leafl, whole common difference, will be unit fimpls equations that divide it without a remainder. or a divifor m of the coefficient of the higheft term, and If fuch fimple equations' cannot be found, yet if we which will be the coefficient of x in the fimple divifor recan find the quadratic equations from which the propofed quited: and whofe term, arifing from the iiippofition equation is produced, we may difcover its roots after- of x—o, will be n, the other member of the fimple diwards by the refolution of thefe quadratic equations. vifor mx—71. Or, if neither thefe fimple equations, nor thefe quadra- From which this rule is deduced for difeovering fuch tic equations can be founds yet, by finding & cubic or a fimpie divifor, when there is any. biquadratic that is a divifor of the pfopofed equation, we may deprefs it lower,, and make the folution more Rule. Subftitnte for x in the propofed equation fuceeffively the numbers 1, o, —1. Find all the divieafy. fors of the fums that refult from this fubftftution, and Now, in order to find the rules by which thefe divi- take out all the arithmetical progreffions you can find fors may be difcovered, we fliali fuppofe, that amongft them, whofe difference is unit, or fome divitnx—n (*fimple of the coefficient of the higheft term of the equamx1—nxAgr C are the < quadratic » for tion. Then fuppofe 71 equal to that term of any one mx"*—wx^-ffx—rj (cubic . progreffion that arifes from the fuppofition of,x’—o, m— the forefaid divifor of the coefficient of the divifors of the propofed equation; and if E reprefent and term of the equation, which m is alfo the difthe quotient arifing by dividing the propofed equation by higheft ference of the terms of this progruffion; fo fhall you that divifor, then have rnx—11 for the divifor required. You may find arithmetical progreffions giving divifors Eyhn . *—jix/t. will not fucceed; but if there is any divifor, it will Or, EXmx'1—nx'/rx—s, will reprefent the. propofed that thus by means of thefe arithmetical prqgreffions. equation itfelf. here it is plain, that “ fince is the be Iffound equation propofed has the coefficient of its ,,“ coefficient of the higheft term of the divifors, it muft highefttheterm==i, it will be w=i, and the divifor be a divifor of the coefficient of the Kigheft term of will be x—n, and then the rule will coincide with that given ' the propofed equation.” in the end of the laft chapter, which we demonftj ated Next we are to obferve, that, fuppofing the equation after manner; for the divifor being x—n, the has a fimple divifor mx—«, if we.fubiHtute in the equa- value aofdifferent be - n, the term of the progreffion that tion EXmx—», in place of x, any quantity, as a, then is a diviforx will that arifes from fuppofing x=o. the quantity that will refult from this fubflitution will Of this cafeofwethegavefumexamples the laft chapter; and necefiarily have ma—n f r one of its divifors: fince, in though it is eafy to reduce aninequation whofe higheft: .this fubflitution, mx—« becomes 771a—n. • term has a coefficient different from unit, to-one where If. we fubflitute fucceffively for x, any arithmetical coefficient ffiall be unit, by p. ;o6. col. 1. par. 6. ; progreffion, a, a—e, a—2e, &c. the quantities that will that without that redu&ion, the equation may be refolved refult from thefe fublUtutions will have among their di- yet, by this rule, as in the following vifors Examp.