Roots of Equations . 
371 
PROP. I. 
To find a common measure of the quantities ax n - bx nm ~ 4 
~f~ cx n ~ 2 -j- d x n —z -j- and Ax n ~ l -f- B x n ~~^ ~j~ C x n ~z 
-j- -f- 
In order to avoid fractions, multiply every term of the divi- 
dend by A 2 , the square of the coefficient of the first term of 
the divisor, and the operation will be as follows : 
A ^c~;+^ 7 > A2 x n -{- bA z x n — i^-chSxn — 2 — j- d A‘ l xn — 3-j- &c. ^A.r-f &A— 
aA' 2 ‘x n -\-aA)Axn — 1 -\-aCAxn — 2 -\-aDAxn — 3-[- 
* [bA 1 — (iBA)^— i-J- (cA a — aCA)xn— 2-f [dA 7 , — «DA);r» — 3 + ^* 
( bA z — a 2 >A)xn — i-f (£BA — a¥> l )xn — 2-f (&CA — «BC)^k — 3 - 1 -^c* 
(P) (cA 1 — 6 BA-f<z] 3 z — aCA) xn — 2-f 
(d A z — b CA-f a BC — a DA) x * — 3 + 
Let (rA~6B) A + (B 2 — CA ) * = * 
(dA— b C) A + (BC — DA) a = ( 3 
(eA— 6D)A+ (BD — EA) a = y &c. 
and the first remainder (P) is ccx n ~ 2 + (3 x n ~ 3 -f* y x*~± + &c . 
proceed with this as a new divisor, and the next remainder (Q) 
will be (Ca — B/3. a -j- / 3 2 — uy . A) x n "~z -J- (D « B y „ 
a -j- @y — . A) .r” - ^ -{“ &c. 
Respecting this operation we may observe : 
l. That were not every term of the first dividend multiplied 
by A 2 , that quantity would be introduced by reducing the 
terms of the remainder (P) to a common denominator. 
q. When P — o, Ax"- 1 -{- Bx n ~ 2 -{-Cx n ~3 &c. is a di- 
visor of A 2 (ax n + bx n ~ x + cx n ~ 2 + &c .) ; and therefore it is 
a divisor of ax n i- bx n_1 + cxn ~ 2 + & c - un * ess it be -a divisor 
gBs 
