OPERATIVE DIVISION 233 



the last equation by — 2 ; which gives the transformation 

 o = 1 — yy + 6y s + y s and the root #3 = 3-12842 . . . 



C. = 2 + io# 4 — x*. 



To transfer the origin by dividing by — 10 would give a 



long transformed equation ; but by putting x = -, we have 



= 1 — iqy — 2y l ; which quickly gives the only real root 

 x = io'oooi9998 . . . 



D. o = 7 — yx + x l . 



Dividing by 7, we see by (2) that, as -x 3 is only a little less 



than ~x', there must be two real positive roots close together 



and that the series for the lesser one will be very slowly 

 convergent. In fact we have the series x= 1 + * 1428 + '06 12 + 

 '03 50 + '0229 + etc., and are still far from the root. Proceed 

 at first, therefore, by Horner's process, using only operative 

 division, as an exercise, for the successive transfers of origin. For 

 the first step divide o s — 70 + 7 by — 1 , giving o $ + 30" — 40 4 1 ; 



and operate with this quotient on — and then multiply 



throughout by 1000 : and proceed as shown below : 



[ioooo][o J — 70 + 7][o 4 1] — = o s + 300* — 4000 4 1000 



[ioooo][o* 4 300 8 — 4000 + iooo][o + 3] — = o s + 3900* — 



193000 + 97000 

 [ioooo][o' + 3900 s — 193000 + 97ooo][0 + 5] — =■= o' + 



+ 40500* — 15325000 + 10375000 

 [ioooo][o : + 40500 1 — 15325000 + io375ooo][o + 6] — = 



= o s + 406800 2 — 1483892000 + 1 325416000 



Here of course operation upon a factor such as 4- 5 is 

 equivalent to division by that factor reversed, that is by — 5, 

 which = [0 + 5] -1 . At any step we may arrest Horner's 

 process and return to operative division. We then take the 

 selected quotient, as for instance after division by — 5, and 



