ALGEBRA. joi 
quadratics, is the fum of two r oots, its diffe rent values 
are a-\-b, a+c, b+c, — a-\-h , — a -jrf, — b+c, and the va- 
lues of e\ or^, are a+21% a+c}% all of which 
bein'? poffible, the cubic cannot be folved by any diredt 
method. Suppofe the roots of the biquadratic to be 
a—b • / i , —a+ c f— i, — a—c f — i; tlie 
values of rare 2 a, b-\-c.f —1, b — c. f — 1, — b — c.f —1, 
[, y T f 1, and —2 a\ and the three values of 
•v are, 2*]', — b+P*, —b—c 1% which are all poffible, as 
in the’preceding cafe. But if the roots of the biquadra¬ 
tic be a-\-bf — t, a — b y/ — 1, — a-\-c , — a—c , the values 
of y are Ia]\ c-\-byf ^\\, c—bf — 1', two of which are 
impoffible; therefore the cubic may be folved by Cardan’s 
rule. 
*r 
Dr. Hearing's Solution of a Biquadratic. 
Let the propofed biquadratic be x i -f2px’ s —qx'-\-rx-\-s ; 
now if there-- 
fore p 2 -\-2Ji.x 2 4-2 pnx-\-n* be added to both fides of the 
propofed biquadratic, the firft par t is a com plete fquare , 
fjfpffnf, and the latter part, 2.pn -\-r 
is a complete fquare, if yxp'f-zn-^qyCrffs =2 
2pn-\-rY ; that is, multiplying and arranging th e terms ac- 
cording to the dimenfions of n, if 8fl’ , 4-4<7«'‘4'8- s —4 rp . n 
_|_4jj^_4^h—r“—o. From this equation let a value of n 
be obtained and fubftituted in the equation at -\-p w-j -nY — 
p*jy 2 n-yq. x t -\-2pnJ r r.x-\-n 2 f-s, then extracting the fquare 
root on both fides, x 2 4-/’*+” = ±:fp~-\-zn-\-q.x-y. f if -\-s 
when 2 pn r is poiitive, or x 2 4 - px n — ±. 
p' l -\-2v-\-q.x — f ri‘-\-s, when 2 pn-\-r is negative, and 
from thele two quadratics the four roots of the given bi¬ 
quadratic may be determined. 
Ex. Let x *—6x 3 4-3x 2 4-2x—10=0 be the propofed 
equation. By comparing this with the equation x^-fzpx 3 
~—qx* — rx — s= o, we have zp ——6 or p— —3, q— —5, r— 
—2,-5=10 ; and — yrp.n^-yqsJ^-yp^s —r 2 = o, 
is Sn 3 —20?i 2 -{-56tz-hi56=:0, or 2 n 3 —3^4-14724-39=0, 
3 3 l 
one of whofe roots is-; hence x 2 —yx -= x'4-7x 
2 2 
4-—, and x* —3a-—-=±: x4--> or x*— 4.x —c=o, and 
* 4 2 2 
,* = .—2x4-2=05 the roots of thefe quadratics,—1, 5, i-J- 
f — x> j— f —1, are the roots of the propofed biquad¬ 
ratic. 
This folution can only be applied to thofe cafes in which 
two roots of the biquadratic are poffible, and two impoffi¬ 
ble. Let the roots be a, b, c, d, then n—f if-\-s, the laft 
term of one of the quadratics to which the equation is re¬ 
duced, is the product of two of them, as ab\ therefore, 
n — ab—f if -\-s, and, fquaring both Ikies, rf — inab^-afb 3 
•—k-fr-s, or •—2 nabJyaffr^as —— abed, and dividing both 
fides by — ab, 2 n — ab—cd, or m—abf-cd , and x= 
abJ^-cd 
the other values of n are 
acfbd 
and 
s d-\-bc 
2 22 
therefore, when a, b , c, d, are poflible, the values of n are 
poffible. Alfo, when thefe quantities are all impoffible, 
the values of n are all poffible ; in neither cafe, therefore, 
can the value of n be obtained by Cardan’s rule; but, if 
two roots of the biquadratic be poffible, and two impofli- 
ble, two values of n will be impoffible, and thexubic may 
be folved; and consequently the roots of the propofed 
equation may be found. 
Voi,. I. No. 19. 
The Method of Divifors. 
Since the laft term of an equation is the prodtnfl of all 
the roots with their figns changed, if any root be a whole 
number it muff: be found among!!: the divifors of the laft 
term. 
Ex. Suppofe x 3 —4-v*—6x4-12=0; the-divifors of the 
laft: term arc 1, — r, 2, •—2, 3, — 3, 4, —4> 6> — 6, 12, 
—12; and, by fubftituting thefe fucceffively for x, we find 
that —2 is a root of the equation. 
When the laft term admits of many divifors, the num¬ 
ber of trials may be leflened by finding the limits between 
which the roots of the equation lie; or, by increaling or 
diminilhing the roots of the equation, and thus leffening 
the number of divifors of the laft term. 
The number of trials may alfo be leffened by fubftitu¬ 
ting three or more terms of the arithmetical progreffion 
1, o, —1, &c. for the unknown quantity, and forming the 
divifors of the refults, taken in order, into arithmetical 
progreffions, in which the common difference is unity; as 
it will only be neceffary to try thofe divifors of the laft 
term of the equation which are terms in thefe progreffions. 
Let x-j-a.f?—o be the equation, one fadtor of which is 
.r-j-a, and Q the product of the reft; if 1,0, —1, be fuc-s 
ceffively fubftituted for x, th.e refults are refpedfively di- 
vilible by 2*4-1, a, and a —1; therefore amongft the divi¬ 
fors of the refults, formed into arithmetical progreffions 
in which the common difference is unity, is found the de- 
crealing progreflion a-J-i, a, a —1 ; and, if all the terms 
correfponding to a , with their figns changed, be fubftitu¬ 
ted in the equation for x, the integral values of x will be 
difeovered. 
Ex. Let the propofed equation be x 3 —qx 2 —6x4-12=0', 
Supp. 
Refults. 
Divifors. 
Progt. 
3 
3 
3 
x= 0 
I 1 
i) 2 » 3 > 4 . 6 . 12 
2 
x ——1 1 
13 
1. 13 
I 
The only decreafing progreffion that can be fonffed out 
of the divifors is 3, 2, 1 ; therefore if one root of the 
equation be a whole number, —2 is that rapt, and on trial 
it is found to fucceed. 
If the higheft power of the unknown quantity be affect¬ 
ed with a coefficient, let wx-J-aXjfl— 0 he the equation, 
and fubftitnte 1, o, —1, fucceffively forx, then«4- w J 
and a—m are divifors of the refults, if the equation have 
a factor of the form mx-X-a, or a root-Alfo m. the 
m 
common difference in the arithmetical progreffion af-m, a, 
a — m, isadiviforof the coefficient of thefirfttermof the 
equation. In this cafe, therefore, all the decreafing pro¬ 
greffions muft be taken, out of the divifors of the refus¬ 
ing quantities, in which the common difference is unity, 
or lome divifor of the coefficient of the higheft term of 
the equation, and amongft them is the progreffion 
a, a — m ; therefore, by making trial of the progreffions 
thus obtained, fucceflively, the fadtor rnxf-a, which di¬ 
vides the equation without remainder, will tie found. 
Ex. To find a divifor of the equation 8x 3 -—26 x 2 4-iix 4- 
10=0, if it admit one of the form mx-' r a. 
Sup. ;Ref.| Divifors. | Progrefs. 
x= i| 3 |i> 3 > I >" 3 
X= 0 JO!I,2,5, 10,— 1 ,—-2,— 3 ,—10, 
3 ) 3 > 3 
^ 2 , —5 
x — - I 35 ! *5 7 > 3 :>> 1 ’ 5 > /> 55 ) 
1 > 1 } 1 
The decreafing progreffions, in which the common dif¬ 
ference is a divifor of 8, formed out of the divifors, are 
3, i) —i-5 3, 2, 1 ! and—3,—5,—7; therefore the fac¬ 
tors to be tried are 2x4-1, x-j-z, and ix —5, the laft of 
which fuccecds, and confequently 2x—3= 0, or x= 
5 
2 
3H 
If 
