i 33

A Treatise of Algebra.x —4.) jr3-{-4* 2 —17X—60 (x 2 -^-8x-^-l5

* 3 —’ 4* 2 _

_ -f8* 2 - l-J X

-4- 8 a - 2 -3 2 X

-j-15*—60-j-15.*-—60o o

*+3) a 2 +8 *+is (*+5**-H ¥ _

. +5- ¥ + I S

+ 5 *+i 5o o

As another Instance hereof, let there be proposed theEquation 2*3 — 3.x 2 -{- r6* — 24. = o 3 then by ex-pounding x by 2, 1, o and — 1 successively, and pro-ceeding as in the foregoing Examples, the Work willstand thus:

2

+ 12

r. 2 . 3.4. 6. 12

+ 2

— r

K

— 9

1.3.9

+ 3

+ l

O

— 24

t. 2.3.4. 6. 8 &c.

+ 4

+ 3 *

I

— 45

1.3.5.9.15.45

+ 5

+ 5

Therefore the Quantities to be tried, in this Cafe,being 4. and 1 ; I first attempt the Division by a-— 4,but find it does not answer; but trying x — [ or (itsDouble) 2x —3, I find it to succeed ; the Quotientbeing * 2 -f- 8.

Nate. The Reason why the Divisors thus found donot always succeed, is, because the first Progression 2,1,o, — 1 is not continued far enough, to know whetherthe corresponding Progression may not break off, after acertain Number of Terms, which it never can whenthe Operation succeeds. Thus, in the last Example, inwhich we had two Progressions 2, 3, 4, 5 and — 1, r,3,5, had the Progression 2,1,0, —1 been continued only

K z twa