ROOT. 



3. Since every equation 



x" + Ax- 4- B*™- 1 + Cx m -~> + &c. + K=. o 



is corapofed of the faftors 



(* - a) (* - /3) (x -y) (x - S) &C. = O, 



it is obvious, that if, by any means, one of thofe roots, a3 a, 

 can be found, the original equation may be divided by \x — a), 

 and thence be reduced to another of lower dimenfions. 



4. If we fuppofe the figns of each of the roots of 

 an equation to be changed, then the co-efficient of the 

 fecond term of that equation will be equal to the fum of all 

 thofe roots fo changed ; the co-efficient of the third term 

 equal to the fum of all the products that can be formed 

 with them, taken two and two at a time ; the co-effi- 

 cient of the fourth term equal to all the produces that can be 

 formed with them, taken three and three at a time ; and io 

 on to the absolute term, which is equal to the produdl 

 of all the roots ; thus, if a, b, e, d, be the roots of an equa- 

 tion, then the co-efficient of the firfl term being 1, that 

 of the 



2d = — (a -f- b 4 c 4 d) 

 3d ^ (ab + ac 4- ad 4 Be 4 bd 4 cd) 

 4th = — {a be 4 abd + acd 4- bed) 

 5th = abed 



c. If the fubltitution of any two numbers, m and n, in- 

 flead of the unknown quantity of an equation, give retults 

 with contrary figns, one, or fome odd number, of the real 

 roots of the equation, are contained between thofe two 

 limits. 



6. And converfely, if two numbers be fubitituted for the 

 unknown quantity of an equation, which cqmprife between 

 them any odd number of the roots of that equation, the 

 refults thus obtained mull neceffarily have contrary figns. 

 But if two, or any even number of roots, be compnftd 

 between thofe limits, then no change of figns will take 

 place in the refults. See a demon llration of thefe properties 

 in Barlow's Tables. 



7. Therefore, when we fubftitute for the unknown quan- 

 tity of an equation the feveral terms of the progreffion, 

 o, I, 2, 3, &c. it will furnilh us with the integral limits of 

 all the real politive roots of that equation, provided it has 

 not any that differ from each other by a quantity lefs than 

 unity, or if it have any odd number of fuch roots ; but if 

 two, or any even number of its roots, be comprifed between 

 two confecutive integers, then thefe fubftitutions will not 

 enable us (at lead not by the change of fign) to difcover 

 the integral limits between which they are comprifed. 



8. But if we fubftitute for the unknown quantity the 

 feveral terms of the progreffion, o, A, 2 A, 3 A, &c. A being 

 fuppofed lefs than the difference of any two of the real roots 

 of the equation, then the limits of every real pofitive root 

 will be indicated by the feveral changes of figns in the re- 

 ipe&ive refults. 



The above two properties have chiefly reference to La- 

 grange's method of approximation. 



9. If, in an equation, whatever real value be fubfti- 

 tuted for the unknown quantity, the refult is always 

 pofitive, it is certain that all the roots of that equation are 

 imaginary. 



10. The figns of all the roots of an equation may be 

 changed from pofitive to negative, or from negative to 

 pofitive, by changing the figns of the alternate co-efficients, 

 •viz. the 2d, 4th, 6th, &c. ; and hence the finding the real 

 roots of an equation is reduced to that of finding pofitive 

 roots only. 



1 1. Every equation ef an odd degree has at lead one rcaJ 

 root, which will be pofitive if the laft term be negative, or 

 negative if that term be pofitive. 



12. Every equation of even dimenfions, having its laft 

 term negative, has at lead two real roots, one pofitive and 

 the other negative ; but if its laft term be politive, it fur- 

 nifhes us with no means of judging of the nature of the 

 roots. 



13. The firfl term of an equation having (as we have 

 fuppofed throughout) unity for its co-efficient, its greateft 

 politive root will be lefs than the greateft negative co- 

 efficient plus 1. 



14. And the abfolute term of an equation being divided 

 by the fum of that term, and thegreatelt co-efficient having 

 a contrary fign, will give a limit lefs than the lead root of 

 that equation. 



15. An equation, having unity for the co-efficient of its 

 firft term, and integral co-efficients for all its others, cannot 

 have a fractional root, w's. its roots muft be either integral, 

 irrational, or imaginary. 



16. An equation cannot have more real pofitive roots, 

 than there are variations in the fucceffion of the figns of its 

 co-efficients, nor more real negative roots than there are 

 permanencies of figns. 



Therefore, when all the roots of an equation are real, 

 there are precifely as many pofitive roots as there are varia- 

 tions, and as many negative roots as there are perma- 

 nencies. 



17. When any term of an equation is wanting, or has its 

 co-efficient equal to zero, and the preceding and following 

 terms have the fame figns, the equation has neceffarily fome 

 imaginary roots. 



18. An equation cannot have all its roots comprifed be- 

 tween two confecutive integers, nor between any two in- 

 tegers, of which the difference is not greater than 2. 



Thefe properties, the demonllrations of which are given 

 by different algebraical authors, may frequently be advan- 

 tageoufly confulted in determining the nature and limits of 

 the roots of equations. 



On the Forms of the Roots of Equations. — We have before 

 obferved, that the roots of equations, beyond thofe of the 

 fourth degree, cannot be generally exhibited in an analytical 

 ferm ; yet from the analogy difcoverable in thofe of in- 

 ferior dimenfions, there feems little doubt but that thofe 

 of the fifth and higher dimenfions partake of the fame 

 form. 



When an equation of the fecond, third, or fourth degree 

 has its fecond term taken away, to reduce it to its molt 

 convenient form for folution, the roots will have the fol- 

 lowing form. 



Seeond Degree. 

 x = a n/p, x — a' iV / p, where a, a', are the roots of 1. 



Third Degree. 



where a and a' are the two imaginary roots of \/ 1. 

 Fourth Degree. 



" = ^P ■•+ V 9 *■ V r 4 where a, a\ a\ are 



= a %f p 4 a -\/ q 4 ^ \f >' 1 the three ima- 



ginary roots of 



.v = a- */ p 4 "' </ q 4 a V 



x = o' \/p 4- a \/ q 4 <? \/ r J 



Whence 



