298 
ALGEBRA. 
quantity greater than a be fubftituted for *, as every faflor 
is, on.this fuppofition, pofitive, tire refult will be pofitive; 
if a quantity lefs than a, but greater than b, be fubftitu- 
tcd, the refult will be negative, becaufe one fadtor will 
be negative and the reft pofitive. If a quantity between 
b and c be fubftituted, the refult will again be pofitive, be- 
caule two fadtors are negative and the reft pofitive, and fo 
on. Thus, quantities which are limits to tire roots of an 
equation, it fubftituted for the unknown quantity, give 
relults alternately pofitive and negative. 
Converfely, if two magnitudes, when fubftituted for 
the unknown quantity, give refults affedted with different 
figns, an odd number of roots nruft lie between them; and 
it a feries of quantities can be found, which give as many 
refults, alternately pofitive and negative, as the equation 
has dimenlionSj thel'e muff be limits'to the roots of the 
equation; becaufe an odd number of roots lies between 
each two (ucceeding terms of the feries, and there are as 
many terras as the equation has dimenfions; therefore this 
odd number cannot exceed one. 
If the refults arifing from the fubftitution of two mag¬ 
nitudes-for the unknown quantity, be both pofitive or 
both negative, either no root of the equation, or an even 
number of roots; lies between them. 
Cor. If m, and every quantity greater than m, when 
fubftituted for the unknown quantity, give pofitive refults, 
m is greater than the greateft root of the' equation. 
To find a limit greater than the greateft root of an equa¬ 
tion. Let the roots of the equation be a, b, c, Sec. tranf- 
form it into one whofe roots are a — e, b — e, c — e, See. and 
if, by trial, fuch a value of e be found, that every term of 
the transformed equation is pofitive, all its roots are ne¬ 
gative, and confequently e is greater than the greateft root 
of the propofed equation. 
Ex. 1. To find a number greater than the greateft root 
of the equation* 3 —5*'4-7-v—xr=o. 
Aflame xx=y- j-e, and we have 
y 2 + 39 , '+ 3 e2 y+ £ ' 3 ] 
— sf— ioey— s? 1 
+ V+l e j 
In which equation, if 3 be fubftituted fore, each of the 
quantities, e 3 —5^+7^—r, 3?—5, is pofitive, 
or all the values of y are negative ; therefore, 3 is greater 
than the greateft value of *. 
Ex - Tm nntr rnliip #»nnof-irm tViic form vL 
=0, which, as far as it goes, has the fame figns with the 
former; and therefore the original equation will have one 
more change of figns, or one mdre continuation of the 
fame fign, than the limiting equation, according as the 
figns of P and Q are different, or the fame. 
Suppofe a, y, Sec. to be the roots of the limiting 
equation; a,b,c,±d, the roots of the original equation; 
then P~- — a x—| 3 X — 7 X &C. and Q~— ax-f>X— fX 
rpr/x Sec. which rectangles will have the fame fign when 
the multiplier d is pofitive, or the root (— d) negative, 
and different figns when that root is pofitive. It appears 
then, that if the original equation have one more change 
of figns than the limiting equation, it has'one more pofi¬ 
tive- root, and, if it have one more continuation of the 
fame fign, it has one more negative root; therefore, if the 
equation Ax n 1 — Sec. —o have as many changes 
of figns as it has pofitive roots, and as many continuations 
of the fame fign as it has negative roots, the fame rule 
will be true in the next fuperior equation .x n — -fix’ 1 ‘4- 
qx n 2 ~- Sec. —o. Now', in every fimple equation, *—a 
~ o, or *4-0=0, the rule is true, therefore it is true in 
every quadratic x'‘±px±q=.o\ and if it be true in every 
quadratic, it is true in every cubic, and fo on; that is, the 
rule is true in all cafes. 
In the demonllration, each root, ±d, is fuppofed to be 
a diftinCt pollible quantity. Hence, when all the roots are 
poftible, the number of pofitive rootsisexadtly known. 
When any coefficient vanifhes, it may be confidered 
either as pofitive or negative, becaufe the value of the 
whole expreffion is the fame on either fuppofition. 
Ex. 1. The equation a- 1 4-*’—14*4-8=10, has two pofi¬ 
tive and one negative root; becaufe the figns are 
—, in which there are two changes, one from -j- to 
—, and the other from — to -f-, and one continuation of 
the fign -f-. 
Ex. 2. If the roots of the equation * 3 — qx-\~r —o be 
pofiible, two of them are pofitive and the third negative; 
for there are two changes of figns in the equation * 3 ;±o—> 
?- v + r—o, and one continuation of the fame fign. 
To find between which 1 of the roots of a propofed 
equation, any given number lies. Let the roots of the 
propofed equation be diminifhed by the given number, 
and the number of negative roots in the transformed equa¬ 
tion will fhew its place among the roots of the original 
equation. 
In any cubic equation of this form, 
=0, f q is greater than the greateft root. 
By transforming the equation, as before, 
/-r 3 f /+ 3 Q'+ 
“ v- 
>■+ 
y—ae V=:0. 
+ ' J 
and, fubftituting \/q for e, jy 3 4~3t/ t }y 1J r i qy-b r —°i eve * 
ry term of which is pofitive; therefore, q is greater 
than the greateft value of *. 
If the equation have both pofitive and negative 
roots, and —-d be the lead root; when d is greater than 
the greateft root, it ie greater than 3/ 
Ex. To find between which of the roots of the equation 
* 3 — 9*“4-23*—.15—0, the number 2 lies. 
A flume *=)’-f 2, then, 
y' , -\-6y‘-\-i2y-\- 8"j 
—9/— 3 fy ~36 l __ 0 
4 - 23 .y+ 4 6 j 5 
— 15 J 
or y J — iy z — -yf^—o. which has one negative root; and 
the roots of the propofed equation are all pofitive; there¬ 
fore two of them are greater, and one lefs, than 2. 
In general, the iaft term and the coefficients of the other 
terms of the transformed equation are found by fubftitu- 
ting the number, by which the roots are to be diminifhed, 
for *, in the quantities, 
\f^7q 
or 
q 2 -~-2pr-\~2s 
-px n 
‘4 -qx r 2 — &c. 
— i.px 11 2 4 -n —2 
qx‘ 
~px n 3 4 -n- 
Sec. 
— n — 
-qx 
— Sec. 
The original equation has as many pofitive roots, and as 
many negative, as the limiting equation, and one more, 
which will be pofitive or negative according to the nature 
of the equation. 
Every equation whofe roots are pofiible, has as rrf’any 
changes of figns from 4- t0 —> and from — to 4 *> as 
has pofitive roots; and as many continuations of the fame 
fign, from 4- to’ -{-, and from — to —, as it has negative 
roots.. Let x n — px n 1 .... drSx® ±:Px^zQ—o, the equa¬ 
tion of limits is nx n 1 .px n ~~ t . . . ±:2.Sx±:P 
And by fubftituting, fucceffively, different numbers for 
*, the limits of the roots of the propofed equation may 
be found. 
On the DepreJJton and Solution of Equations. 
If an equation contain equal roots, thefemay be found, 
and the. equation reduced as many dimenfions lower as 
there are equal roots. Let the roots of the equation . 
px rL ~'-\-qx n 2 -— Sec. ~o> be a,b, c , d, Sec. then, 
nx r ’■ *. 
