OPERATIVE DIVISION 395 



division. For this purpose divide algebraically by x and take 

 the " setting " 



x 1 + x~ l = 5. 



Then o : 



Thus # 



= ±2*2361 — o* 1000 ^f 0*0067 

 = 2-1294 or — 2-3293 



The actual roots are x = 2*1284 • • • an d — 2*3300 . . . ; so 

 that, in this case, with only three terms of one invert we have 

 obtained three correct figures for the first root and nearly three 

 correct figures for the negative one. 



The middle root, x = 0*201637, . . . can be obtained by 

 ascending division from the setting $x — x z = 1 , as already given 



in Section III (5). Or, putting x — - and then dividing by z\ 



we get the setting s 2 + z~ l = 5, which yields the same root by 

 descending division. The reader may obtain it and also ex- 

 plain the facts. 



C. x 3 — 3# 2 — 2x + 5 =0 



This has the same roots as the previous example, but increased 

 by unity. The setting x — 2x~ l + $x~ 2 = 3 gives the greatest 



root by descending division, but slowly. Putting x = -, 



z 



we have z 3 — 2s 2 — 15s + 25 = o ; and from the setting 

 z* — 2z + 25s" 1 = 15, we obtain from one invert, but slowly, 

 z = 4*237 . . . and = — 3*901 . . ., which, when the substitution 

 is made good, approach the negative and the middle roots of x. 

 The setting x 2 — 3X + 5# -1 = 2 gives two series (for the two 

 roots of \/2) which converge very slowly if at all towards the 

 greatest and least roots ; and the centred form of the previous 

 example is evidently preferable. 



