46 Algebra. 



Now it is evident that, keeping the first sign unchanged, there 

 is no other arrangement of signs than those written in this scheme, 

 except the arrangement H- + + , which is the arrangement of the 

 given trinomial. Therefore 



The ?'ationalizing' factor for a?iy bdnornirial quadratic surd is the 

 product of all the different trinomials which ca7i be made from the 

 original by keeping the first term imchanged ajid giving the sig?is 

 4" (ind — to all the reinaijmig terms i7i every possible order, except 

 the order occurri?tg i7i the give?t trinomial. 



As an example, find the rationalizing factor for ^^5— "^Z+^J. 

 The above method shows it to be 



and multiplying the original trinomial by this the rationalized re- 

 sult is found to be —40. 



The above problem is cabable of generalization, but its proof 

 cannot be practically given here. The generalized statement is 

 as follows: 



The rationalizing factor for any polynomial quadratic surd is 

 the product of all the differe7it poly7iomials which can be made from 

 the origi7ial by keepi7ig the first ter77i imchanged a7id givi7ig the 

 sig7is -j- a7id — to all the re7nai7ii7ig te7i7is i7i every possible 07'der 

 except the order occurring i7i the give7i polyno7nial. 



26. Problem. To Rationalize any Binomial Surd. Abi- 



h p b p 



nomonial surd will either take the form a^r-^c~q or a '-—c'^. Now 

 since these fractional exponents may be reduced to a common de- 



nominator SO that the expressions become a^<!-\-c''J or a'-^—C'^ these 



1 L St 



binominal surds may be supposed in the form a''-\- c or a" — c'\ 



These, then, are the only forms necessary to consider, since all 



binomial surds are reducible thereto. 



(a) To 7'atio7ialize the fo7in a" — c" . 



.s- / .V / 



For convenience let a"=x and r"=j';when a" — c" =^x—y. 

 Now multiply x—y by 



x"-'-\-x"-y+x"-Y+ . . .y-' (i) 



and we obtain 



