z 9 S ALGEBRA. 
o ily one of the quantities, a, this equation is a 71 — pa n ' 
-\-qa n *■— &c. =o, which exactly coincides with the 
propofed equation ; in fuppofing, therefore, that a can be 
found, we take for granted the propoiition to be proved. 
The fubjeifl has exercifed the (kill of the moft eminent 
algebraical writers, but their reafonings upon it are of 
too a'oftrufe a nature to be introduced in this place. The 
learner mull, at prefent, take for granted, that an equa¬ 
tion may be made up of as many fimple faftors as it has 
dimenfions. 
The quantities, a, b, c, d, &c. are called roots of the 
equation, or values of x, b'ecaufe, if any one of them be 
fubftituted for a, the w'hole expreflion becomes nothing, 
which is the only condition propofed by the equation. 
Every equation has as many roots as it has dimenfions. 
If a" — px n '-\-qx n 2 — Sec. e=o, or a— a.x — b.x — c. 
Sec. to n factors' 2=0; there are n quantities, a , b , c, Sec. 
each of which, when fubftituted for x, makes the whole 
=:o; but any quantity different from thefe, as r, when 
fubftituted for x, gives the product e — a. e—b.e — c. Sec. 
which does not varii'fh, that is, e will not anfwer the condi¬ 
tion which the equation requires. 
When one of the roots, a, is obtained, the equation 
x — a.x — b.x — c. Sec. =0, or a" — px n ' '-\-qx n 2 Sec. 
=o, is divifible by x — a, without a remainder, and is thus 
reducible to .v— b.x — c. &c. —o, an equation one dimen- 
iion lower, whole roots are b, c, Sec. 
Ex. One root of the equation y 3 -\-t~o, is —i, and the 
equation may be deprefled to a quadratic, in the following 
manner: 
T+0/+i (/—y-f i 
- 7 ' 
—f—y 
+y + 1 
y+i 
-* 
Hence the other two roots are the roots of the quadratic 
o, . 
If two roots a and b be obtained, the equation is divi¬ 
fible by .*•— a.x — b\ and thus it may be reduced two di¬ 
menfions lower. 
Ex. Two roots of the equation a 6 —1=0, are + 1 an d 
—1, therefore it may be deprefled to a biquadratic by di¬ 
viding by .v—1. x 1, or by a 2 —r. 
A 6 - X* 
x°-\- [ 
* 
Hence the equation a^-I-a 2 -!-1=2:0, contains the other, 
four roots of the propofed equation. 
Converfely, if the equation be divifible by a— a, with¬ 
out a remainder, a is a root; if by x — a.x — b, a and b are 
both roots, See. Let Q be the quotient ariling from the 
divifton, then the equation is a— a.x — b.Q— o, in which, 
if a or b be fubftituted for a, the whole vanifhes. 
Cor. i. If a, b, c, Sec. be the roots of an equation, that 
equation is a— a.x — b.x — c. Sec. =0. Thus the equa¬ 
tion, whofe roots are, 1, 2, 3, 4, is x — i.x — z.x —3 x —4 
■zx.0, or a*—ioa' 3 -j- 35* 2 —jox-f-aqmo. The equation 
w hofe roots are 1, 2, and —3, is .v— i.x — 2-. a-}-3=:o, or x 3 
—qxf-6 —of 
Cor. 2. If the laft term of an equation vanifti, it is of 
the form a”— px n : - \-qx n *. P. x=o, which is divi¬ 
fible by x, or x—o, without remainder, therefore 0 is one 
3 
of its roots; if the two laft terms vanifli, it is divifible by 
w 2 'without remainder, or by a— o.x — 0, that is, two of 
its roots are 0, See. 
The coefficient of the fecond term of an equation, is 
the fum of the roots with their figns changed; the coeffi¬ 
cient of the third term, is the fun; of the products of eve¬ 
ry two roots, with their figns changed; the coefficient of 
the fourth term, the fum of the products of every three 
roots, with their figns changed, &c. and the laft term is 
the product of all the roots, with their figns changed. 
Cor. If the roots be all politive, thelignscf the terms 
will be alternately -f ar >d —For the product of an odd 
number of negative quantities is negative, and of an even 
number, pofitive. But, if the roots be all negative, the 
ligns of all the terms will be pofitive, becaufe the equation 
arifes from the multiplication of the politive quantities, 
a*—• A" b.Sec. 
The roots, a, b , e, Sec. of an equation are impojfible , 
when, as is frequently the cafe, they involve the fquare 
root of a negative quantity. 
Impoftible roots enter equations by pairs. If a-\-f — b 2 
be a root of the equation a" — px 11 Sec. =0, a —• 
3/ —b 2 is alfo a root. 
For .v in the equation, fubftitute a —i 2 , and the re - 
fult will conlift of two parts, poflible quantities, which in¬ 
volve the powers of a and the even powers of 3/ — b 2 , 
and impoflible quantities which-involve the odd powers of 
y — b 2 ; call the fum of the poflible quantities A, and of 
the impoflible quantities B, then AfB is the whole re- 
fult. Let now, a —3/ — ~b~, be fubftituted for a, and the 
poflible part of the refult will be the fame as before, and 
the impoflible part which arifes from the odd powers of 
•—3/— b 2 , will only differ from the former impoflible part 
by its ftgn ; therefore the refult is A—B ; and fince by the 
fuppofition 3/— b 2 is a root of the equation, A-\-B— o, 
in which, as no part of A can be delfroyedby B, A~o and 
B— o, therefore A — B— o; that is, the refult, arifing from 
the fubftitution of a —3/— b 2 for a, isnothin'g, or a —3/— b 3 
is a root of the equation. 
Cor. i. Hence it follows that an equation of an odd 
number of dimenfions muft have, at leaft, one poflible 
root, unlefs fome of the coefficients are impoflible; in 
which cafe the equation may have an odd number of im¬ 
poflible roots. 
Cor. 2. By the fame mode of reafoning it appears, 
that, when the coefficients are rational, lin’d roots of the 
form -±.fb, or aA~\Jb, enter equations by pairs. 
On the. Transformation of Equations. 
If the figns of all the terms in an equation be changed, 
its roots are not altered. Let a — a.x — b.x — c. Sec. =20 ; 
then — a — a.x — b.x —c.—o, when x~a, b, c, See. 
If the figns of the alternate terms, beginning with the 
fecond, be changed, the figns of ail the roots are changed. 
Let x n — px n 1 — qx n 2 -f- Sec. ~o, be an equation whofe 
roots are a, b, —c, &c. for a fubftitute— y, and, when n is 
an even number, y"-\-py n 1 ~—qy n ’ 2 — Sec. =0; but 
when n is an odd number, —-y — py n lJ r<jy n 2 -r & c - 
=0 ; or, changing all the figns, y nj cpy n ■'— qy n 2 —■ Sec. 
—o, as before; and fince X—~-y, ox y— — x, the values of 
y are, •—a , — b, -|-c, &c. 
Ex. Let it be required to change the figns of the roots 
of the equation, a 3 — qx-^rzxeo. This equation with all its ■ 
terms isA 3 -j-o— qxfr—o, and, changing the figns of the 
alternate terms, we have x 3 —o— qx — -r-xzio, or a 3 — q.y—r 
=0, an equation whofe roots differ from the roots of the 
former only in their figns. 
To transform an equation into one whofe roots are great- . 
er or lefs than the correfponding roots of the original 
equation by any given quantity. Let the roots of the 
equation 
