Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/135

 I03 A. B li A L G E When therefore any equation is propofed to be refol&c. =o. will exprefs the equation to be proved, it is eafy to find the fum of the roots, (for it is e- duced ; all whofe terms will plainly be pofitive ; fo that, qual to the coefficient of the fecond term having its fign “ when all the roots of• an equation are negative, it is changed); or to find the fum of the produfts that can “ plain there will be no changes in the figns of the terms be made by multiplying any determinate number of “ of that equation.” them. general, “ there are as many pofitive roots in any eBut it 'is alfo eafy “ to find the fum of the fquares, or “ Inquation as there are changes in the figns of the terms “ of any powers, of the roots. from + to —, or from — to +; and the remaining The fum of the fquares is always />’'—2q. For call- “ roots are negative.” The rule is general, if the iming the fum of the fquares-Z?, fince the fum of the roots pofiible roots be allowed to be either pofitive or negative; is p; and “ the fquare of the fum of any quantities is and may be extended to all kinds of equations. “ always equal to the fum of their fquares added to In quadratic equations, the two roots are either both “ double the products that can beimade by multiplying pofitive, as in this2 “ any two'of them,” therefore p —B--7.g, and confe- (.v—rfX.v—b—) x —ax+ab—o, —bx quently Z?=:/>z —2y. For example, c'-Sriab+T.ac+'ibe; thatis,/>2=-5-|-2y. Anda+H-Hr^’ both wherenegative, there areastwoin this changes of the figns: Or they are that is, again, p'—B+iq, or B—pz—2?. And fofor any other {x+«Xx-M=) x2+^ x+^_0} number of quantities. In general therefore, “ B the “ fum of the fquares of the2 roots may always be found where there is not any change of the figns: Or there is. “ by fubtra&ing, 2q from /> ;” the quantities p and q one pohtive and one negative, as in being always known, fince they are the coefficients in the propofed equation. “ The fum of the cubes of the roots of any equation is equal to />3—or f° Bp—/,y+3^.” For where there is necefiarily one change of the figns; be~B qXp gives always the excefs of the fum of the cubes caufe the firft term is politive, and the kit negative, and of any. quantities above the triple fum of the produfts there can be but one change whether the 2d term be + that can2 be 2made2 by fnultiplying any three of them, Therefore the rule given in the lafl: paragraph extends Thus, tf -H—^abc. -f-t —ab—ac—bcY.a--b--c — qX.p)-=. Therefore if the fum{=B of the cubes to all quadratic equations. equations, the roots may be, is called <7, then ffiaH1 B—q'Ap—C—^r, and C—Bp—qp Ini°.cubic 3 All pofitive, as in this-, x—aXx — bXx—c^=ot (becaufe B—p —2y)=/> —Ipq+Yip which the' figns are alternately + and —, as appears After the fame manner, if D be the fum of the 4th and there are three changes of the figns. powers of the roots, you will find that D—pC—qB+pr from2°.theThetable; may be all negative, as in the equation —4/, and if E be the fum of the 5th powers, then ffiall x-^-aXx-^-bXxroots -+-c=o, E—pD—qC+rB—pj+qt- And after the fame manner the figns. Or, where there can be no change of the fum of any powers of the roots may be found; the 3°. There may fee two pofitive roots and one negative, progreffion of thefe expreffions of the fum of the powers as in the.equation x—a>'x—bXx-{-c=:o; which gives being obvious. As for the figns of the terms of the equation produced, it appears, from infpeftion, that the.figns of all. the terms in any equation in the table are alternately + and . —: thefe equations are generated by multiplying continually x—a, x—b, x—c, x—d, &c. by one another. Here there mull be two changes of the figns; becaufe if The firlt term is always fome pure power of x, and is a--b is greater than c, the fecond term mu ft be negative, pofitive; the*fecond is a potver of x multiplied by the its coefficient being —a—b-{-c. quantities —a, —b, >—r, &c. And fince thefe are all And if a+b is lefs than c, then the third term muft negative, that term mult therefore be negative. The be negative, its coefficient +ab—ac-—■lc{ab—cXa+b) * third term has the produds of any two of thefe quantiin that cafe negative. And there cannot poffibly be ties (—a, —b, —c, See.) for its coefficient; which pro- being changes of the figns, the firft and laft terms having duZtS are all pofitive, becaufe —X— gives +. Far the three the 0fame fign. like reafon, the next coefficient, confiiting of ail the 4 . There may_be one pofitive root and two negative, products made by multiplying any three of thefe quanti- as in the equation x-^raXx--bXx—c—o> which gives ties mull be negative, and the next pofitive. So mat +*nV x-—abc~Q, the coefficients, in this cafe, will be pofitive and negative -f/’ V x1—ac by turns. But, “ in this cafe the roots are all pofitive fince x=a, x=b, x=c, x=d, x—e. See. are the afiumed where fimple equations. It is plain then, that “ when all the “ roots are pofitive, the figns are alternately -j- and —-d* * Becaufe the redangle aXb is lefs than the fquare But if the roots are all negative, then x+aX^+^X a-t-bXa-j-b, and therefore much lefs than a-fbXc.
 * 3-bo