1 62 Algebra. 



14. It is interesting to note that the development of an irra- 

 tional expression may turn out to be a series of a limited number 

 of terms. 



Suppose, for example, we wish to develope \/ i—2x-\-x^- and 

 do not recognize that 1 — 2:1-}--^'' is a perfect square, then assume 

 as before 



s/ i — 2x-^x^=A^-{-Ax-\-AX^-\- . . . 

 Square both sides and we get 

 i — 2x—x'=A^,+ 2\Ax-\- (A; + 2 A^AJa'" 



■ +(2A>3-f 2A A>"'H- .... 

 Bquating coefficients of like powers of x and we get 



A„=i, 

 2AA=-2 .-. A=-i, 

 A;+2A„A^=i . . A^=o, 

 2A^A^-f 2A A^=o .-. A^=o, 

 A/+2A A^+2A A^=o .-. A^=o, 

 etc., etc., 



and each of the subsequent coefficients will turn out to be zero, 

 hence we ^et 



s/ l — 2X-\-X''=^l—X. 



15. In developing irrational expressions it sometimes happens 

 that we should begi7i our assumed development with some negative 

 power of X. 



An inspection of the proposed example will show with what 

 power oi X the development should begin ; for the assumed series 

 must be such that, when the equation obtained by putting the 

 given function equal to the assumed series is reduced to the 

 rational integral form, then the lowest power of x on the side 

 which contains the undetermined coefficients must be as low as 

 the lowest power on the other side of the equation. 



Thus, to develope 1 1 -|- ^ we would begin the assumed series 



with a term containing x~\ for when this is squared the lowest 

 power of X is x~^ and when both sides are multiplied by x"^ 

 to reduce to the integral form then the series on the right side of 

 the equation will begin with an absolute term as it should. 



