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 102 A L G E B R A; whence it is produced. The fquare of an impoflible1 X —12=0 quantity may be real, as the fquare of y'—a* is —a" ; Xx —6=o but “ the cube of an impoffible quantity is ftill impofli- =Xt “ b!e,” as it ftill involves the fquare root; of a negative: ~6x+ah } =°> a as, y”—a* X V—Xy'—a^—^—a6 —a*/—i, is plainly imaginary. From which it appears, that though two fimple equations involving impoffible expreffions, —6^- Xx4+<7c^ Xx—a cubie. multiplied by one another* may give a produdt where no —c ^ --bc J impcffible expreffion may appear; yet “ if three fuch Xx —d=o “ fimple equations be multiplied by each other, the im“ poffible expreffion will not difappear in their produdh” —abe'j And hence it is plain, that though a quadratic equation whofe coefficients are all real may have its two roots im—^yXx+abcd=°, biquapoffible, yet “ a cubic equation whofe coefficients are dratic. •-bc >Xx —bedj real cannot have all its three roots impoffible.” +bd! In general, it appears, that the impoffible expreffions+cdj cannot difappear in the equation produced, but when their number even; that there are never in any equations, whofe coefficients are real quantities, fingle impof+ab -.:bc' -~abcd ^ fible roots, or an odd number of impoffible roots, but +ae •—abd that the roots become impoffible in pairs, and that / Xx4+adj —abe --abce --abdey xx—clcde-C, “ an equation of an odd number of dimenfions has al—acd --acdeJ “ ways one real root.” +bc) —ade “ The roots of equations are either pojitive or negative, +bd X 3 —ace l--bcde “ according as the roots of the fimple equations whence +be —bed “ they are produced are pofitive or negative.” If you —bee {*furfolid. fuppofe x=—ra, x=—b, x——c, x~—d, &c. then ffiall x--,r~o, x--b—o, .v-|-c=o, x+A=o ; and the equation —bde +de) —cdt) a+« Xx-M X*-f cXx + ^ = o will have its roots, &c. •—a, —b, —c, —d. See. negative. But to know when the roots of equations are pofitive, the infpeftion of thefe equations it is plain, that, ‘ and when negative, and how many there are of each theFrom Coefficient of the firft term is unit. kind, ffiall be explained in the next chapter. The coefficient of the fecond term is the fum of all the roots (a, b, c,*d, e,) having theirJigns changed. The coefficient of the third term is the fum of all the Chap. XV. Of the Signs and Coeffi- produfts that can be made by multipying. any two of the cients of Equal, ons. roots (a, b, c, d,' e,) by one another. The coefficient of the fourth term is the fum of all the When any number of fimple equations are multiplied produtts that can be made by multiplying into one anby each other, it is obvious that the higheft dimenfion of, other any three of the roots, with their Jigns changed. the unknown quantity in their produdt is equal to the And after the fame manner all the ether coefficients are numbet of thofe fimple equations - and the term involving formed. the higheft dimenfion is called the firji term of the equa- The laft term is always the produ ft of all the roots hation generated by this multiplication. The term invol- ving their figns changed, multiplied by one another. ving the next dimenfion of the unknown quantity, lefs Although in the table fuch fimple equations only are than the greateft by unit, is called the fccond term of multiplied by one another as have pofitive roots, it is the equation'; the 'term involving the next dimenfion of eafy to fee, that “ the coefficients will be formed acthe unknown quantity, which is lefs than the greateft by “ cording to the fame rule when any of the fimple equatwo, the third term of the equation, <bc. ; and that “ tions have negative roots.” And, in general, if x3 term which involves no dimenfion of the unknown-quan- px^+qx—r=o reprefent any cubic equation, then.ffiall p tity,, but is fome known quantity, is called the lujl term be the fum of the roots ; q the fum of the produfts made by multiplying any two of them; r the produft of all of the equation. “ The number of terms is always greater than the the three : and, if —p, -j-y, —r, -J-r, —t, --u, See. be “ higheft dimenfion of the unknown quantity by unit.” the coefficients of the ad, 3d, 4th, 5th, 6th, 7th, 6r. And when any term is wanting, an afterilk is marked in terms of any equation, then ffiall / be the fum of all the its place. The figns and coefficients of equations will be roots, q the fum of the produfts of any two, r the fum underftood by confidering the following, table, where the of the produfts of any three, s the fum of the produfts fimple equations x-—a, x—h, 8cc. are multiplied by one of any four, t the fum of the produfts of any five, « the another, and produce fucceffively the higher equations. fum of the produfts of any fix, &c. When