ALGEBRA. 
z 99 
__ If & c - 
vx r '■ — n — s.px n ~*-f l J 4 * A ‘— a - x — b.x- — d. 8c c. 
n — i.qx n 3 — 8c c. f | fx — a.x — c.x — d. See. 
J L-J-.v— b.x — c.x — d. &c. 
Suppofe then 
"1 j x — a.x — a.x — c. See. 
— i ,px n ~*-f +y— a - x — a - x — d. See. 
11 X" 
U '—2 
.qx n 3 — &C. I j -f-Y— a - x —c.y- 
mJ Li —l »%" ~'~C • X~~ 
-d. &C. 
__p.,v— a.x — c.x — d. 8cc. 
the whole of which is divifible by a -—a without remain¬ 
der, that is, a is a root of the equation nx n 1 — n —i. 
px K 2 n — 2.qx n 3 — See. =o. 
If three roots, a, b,c, be equal, y— a.x — a. will be 
found in every product; therefore the equation is divifible 
by x—a.x—a. without remainder, or two of its roots are 
a, a. In the fame manner, if the original equation have 
n equal roots, the equation nx n '— n — i.px n 2 -fn —2. 
qx” 3 —&c. =0, has m —1 of thofe roots. Hence it ap¬ 
pears, that when there are m equal roots, the two equa¬ 
tions have a common meafure of the'form x— o) w ‘and 
in roots of the original equation known. Divide this 
-equation by .v—rT| m , and the refulting equation, of n—m 
dimenfions, contains the other roots. 
Ex. Let the equation y 3 — px*-fqx —r—o, have two 
equal roots; then 3Y 2 — 2px-fq—o has one of them, and 
the two equations have a common meafure which is a Am¬ 
ple equation ; alio the quantities 3Y 3 — T,px*-fT,qx —3/-, and 
3Y 2 — 2pxfq, have the fame common meafure, which is 
thus found. 
3 .v 2 —-2/Y -J- j) 3 y 3 — 3/w 2 3 qx — 3 r ( Y - 
3.V 3 — zpx 2 -j- qx 
•—• px 2 fiqx- 
— P x 2 +-t - 
> r 
pq 
6 q —2 p~ <)r—pq 
Hence 
6q—zp* V—pq 
, . 6q — 2p* 
-px 2 -fqx —r=;o; that is, —- x- 
9 r —Pq 
is a divifor of the equation y 3 
9 r —Pq 
By changing the figns of the alternate terms we obtain the 
equation y 4 —jy 3 — qx*-fz.qx — 18=20, which lias two roots 
of the form +*> —« ; their common quadratic divifor is 
y*—9=20, hence Y=nt3- To obtain the other roots, di¬ 
vide y 4 -E3y 3 — 7Y 2 — 27Y — 18=0, by y 2 —92=0, and the roots 
of the refulting equation, x*-f 3v 2 2=0, are the roots 
fought. 
By this method furd roots, of the form def a, maybe 
conftantly difeovered; becaufe they are always of this 
form wdien the coefficients are rational. 
Solution of Recurring Equations. 
The roots of a recurring equation, of an even number 
of dimenfions, exceeding a quadratic, may be found by 
the folution of an equation of half the number of dimen¬ 
fions. Let y”— px n 1 .—^Y-J-i2=o; its roots are 
of the form a, -, b, 8cc. or it may be conceived to be 
a 0 
made up of quadratic factors, x — a.x —-, x — b.x— y 
a b 
See. i. e. if viz 
za -\—, n — b- j—-r, 
a b 
See. of the quadratic 
faftors, y’— mx-fi, y 2 —iiY-j-r, &c. Then, by multi¬ 
plying thefe together, and equating the coefficients with 
thofe of the propofed equation, the values of m, n, Sect 
may be found. Moreover, (ince m is only of one dimen- 
iion, where y is of two, the equation for determining the 
value of m will rife only to half as many dimenfions as.,v 
rifes to in the original equation. 
If the recurring equation be of an odd number of di¬ 
menfions, 4-1, or—1, is a root; and the equation may 
therefore be reduced to one of the fame kind, of an even 
number of dimenfions, by divifion. 
Ex. Let Y 3 —12=0. Unity is one root of this equation, 
and, by dividing y 3 — 1 by y — 1, the equation x* 
—J— A - —|— 1 =20 is obtained, which contains the other two, 
—i + t /—3 , — 1—</— 3 
and y or Y2 
and 
Y —J— Y “j— I 2 
y 2 -|—.v-f- -2 
4 
, 1 
Y -|-222 ± 
2 
-i±V—3 
= 0 
I 
~4 
v~ 
—8 
4 - 
that is, the three roots of the equation 
bq — 2 p 
Thus two roots of the equation are difeovered; and, 
fince r is the produdt of all the roots, r divided by 
or —pq \ 2 
--- I is the third root. 
6q—2p\ 
Let the propofed equation be y 3 — 4 Y 2 -{-jY —22=0. 
:i. If then the 
y 3 —1=0, or the three cube roots of 1, are 1, 
-i-i-y—3 
and 
-i — f —3 
In the fame manner the roots of the equation .v’-j-f—® 
are found to be —1, 
Here p—x, q— 2?, 7-2=2; and --— 
. 6q—2p‘ 
equation have two equal roots, each of them is j, and the 
third root is 2. 
But it mud be obferved, that though 1 be a root of the 
propofed equation, it has not another root alfo 1, unlefs 
1 be a root of the deprelfed equation 3 Y 2 —SY-f 5 2 = 0 . 
If two roots of an equation be of the form -fa , a, 
differing only in their figns, they may be found, and the 
equation deprelfed. Change the figns of the roots, and 
the refulting equation has two roots -fa, — a\ thus we 
have two equations with a common meafure, y 2 — a 2 , 
which may be found, and the equation deprelfed, as in the 
preceding cafe. 
Ex. Required the roots of the equation Y-'-f^Y 3 _ qx* 
—18=0, two of which are of the form 4 -a, —a. 
Solution of a Cubic Equation by Cardan's Rule. 
Let the equation be reduced to the form y 3 — qx-fr=zo } 
where q and r may be pofitive or negative. Alfume y= 
a-fb, then the equation becomes a-fb) 3 — qY.a-fb -fr=o, 
or a 3 -fb 2 -fT,abyta-fb —( 7 X7-22=0; and, fince we 
have two unknown quantities, a and b , and have made on¬ 
ly one fuppofition refpedting them, viz. that a-fb—x, we 
are at liberty to make another; let %ab — q 2220, then the 
equation becomes a 3 -fb 3 -fr=o ; alio, fince 3 ab — q- 2.0, b=z 
q # q 3 
—, and, by fubftitution, a 3 A - 
3# 27 a J 
£- 
2 f 
+r2=o, or a*-fra 3 -f 
o, an equation of a quadratic form: 
and, by com¬ 
pleting 
