Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/143

 BRA. 111 A L G E (a4+£l 4,)6 this mean proportional will be rifes from the propofed cubic by multiplying die terms biquadrates <: l of this latter by the arithmetical progreffion o, —i, —2, 3 '-3-a b*--a*b' +b. And the1 fum of the cubes is .—3. And, in the fame manner, it may be ftiewn that «>/a . Now, fince a*—2ab--b is the fquare o( a—b,1 the roots of the equation px*—iqx'+^rx—4/1=0 are it +£ mull be always pofitive; and if you multiply it by a' b", limits of the equation x*—■/’.v 1 — rx-}-/=o. the produdt —2«3^3H-^1i4 will alfo be pofitive; Or, multiply the1 terms of the equation andJ confequently will be always greater than 3 6 6 2tf i . Add a -~b and we have a6a"'b7---a*b*-^-b^ —px --qx—r=o greater than a6+2a'bl-^-b'•, and extrading the root by a+jt, a+2b, a-^-b, a greater than <i3-H3. And the 1 ax3 —apx -%-aqx—ar (=0)1 _ J fame may be demonftrated of any number of roots what3 1 3^x —2bpx --bqx (=3x —2px rq'Kbx,) ever. Now, if you add the fum of all the cubes taken affirAny arithmetical feries where a is the leaft term, and b to their fum with their proper figns,, they will the common difference, and the products (if you fubfti- 7natively double the fum of the cubes of the affirmative roots. tute for x, fucceffiveiy, a,'bTc, the three roots of the pro- give if you fubtrad the fecond fum from the firft, there pofed cubic) {hall be -j-A^X^x, —M' Xbx+L.Y.bx. For the And iremain double the ium of the cubes of the negative firft part of the product aXx3—px^-Yqx—r—o 1 roots. it follows, that “ half the fum of the <?, b, c, being limits in the equation 3X —2px--q~o, “ mean Whence betwixt the fum of the fquares their fubflitution muff give refults N, M, L, alternately “ and theproportional fum of the biquadrates, and of the fum pofitive and negative. a—I “ of the cubes of the with their proper figns, exIn general, the roots of the equation x” —/>x of the -l- “ ceeds the fum of theroots cubes of the affirmative roots:” q. Xn—t— rx»—t1 -|-} &c.'=:o !are limits of the roots 3 and.‘* half their difference exceeds the fum of the cubes equation «>"— —«—iX/>x' ——2X9X"— —n—3X roots.” And, by extrading the cube, rx"—4-4-» &c. =0: or of any equation that is deduced from ofofthethatnegative fum and difference, you will obtain limits it by multiplying its terms by any arithmetical grogref- root that ffiall exceed the fums of the affirmative and of the fion, az+zb, «z+z2b, ctz+z^b, a-=^z.b, &c. And, c<j«- negative roots. And fincc- it is eafy,. from what has verfely, the roots of this new equation will be limits of been already explained, to diminim the roots of an equathe propofed equation tion fo that they all may become negative but one, it apn pears how, by this means, you may approximate very x"—■px —’-f-yx”—*—, &c. =0. to that root. But this does not ferve when there “ If any roots of the equation of the limits are im- near impoffible roots. poffible, then muff there be feme roots of the propofed areSeveral other rules like thefe might be given for limi**theequation as (inp. no. col. par. 2.)to ting the roots of equations. We {hall give one not menquantityimpoffible.” 3<?1—a/’z-j-yFor was-demohftrated to be1. equal by other3 authors. the produft of the exceffes of two values of x above the tioned In a cubic x —pxz--qx—r=o, find y*—2/r, and call third fuppofed equal to if any irhpoffible expreffion be found in thofe excefles, then there will of coniequence it e*; then {hall 4the£ greateft4 root __ of the equation always be found impoflible cxpreffions in thefe two values of x. be greater than 3, or v/ — • And, And “ from this obfervation rules may be deduced for “ difcovering when there are impoffible roots in equa- In any equationlx"—I+yx'3 1—*—/■x”~3-|-,&c.:=Cj, V tions.” Of which we fhall treat afterwards. Befides the method already explained, there are others and extrading the root of the 4th by which limits may" be determined which the roof of find power out of that quantity, it {hall always be lefs than an equation cannot exceed. .... Since the fquares of all real quantities are affirmative, the greateft root of the equation. it follows, that “ the fum of the fquares of the roots of “ any equation muff be greater than thefquareof the Chap. XIX. Of the Refolution of Equatlons r “ greateif root.” And the fquare root of that fum will all viboje Root are commenfurate. therefore be a limit that muff exceed the greateft root of the equation. 1 3 in chap. 15. that the laft term of If the equation propofed is x”—px"— -pyx” — It was demonftrated is the produd of its roots : from which it rx"—3-p, &c. =0, then the fum of the fquarps of the any equation that the roots of an equation, when commenfu-r roots1 (p. 103. col. 1. par. 1.) will be p1—2y. So that follows, rable quantities, will be found among the divifors of the V'/> —2y will exceed the greateft root of that equation. 1 laft term. And hence we have, for the refolution of, eOr if you find, by p. 103. col. 1. par. 4. the fum of qua tions, this the 4th powers of the roots of the equation, and extraft Rule. Bring all the terms to one fide of1 the equation, ths biquadratic root of that fum, it will alfo exceed the find all the divifors of the laft term; and fubftitute greateft root of the equation. them fucceffiveiy for the unknown quantity in the equaIf you find a mean proportional between the fum of tion. So {hall that divifor which, fubitituted in this the fquares of any two roots, b} and the fum of their maimer.
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