122 Prof. Young on the Evaluation of Vanishing Fractions, 

 2. ~F(x)=x 3 -5x <2 + 3x + 9, f{x)=x 3 -x 2 -2lx + 45, #=3. 



1-5 +3 +9(3 

 3 -6 -9 



-2 -3 

 3 3 





1- 



-1 - 

 3 



2 - 

 3 



-21 +45(3 

 6 —45 



-15 

 15 



1 o 

 3 



4 







5 

 3 



8 







. F 2 (3)_ 



_4_ 



1 







MS) 8 2 



3. 'F(x) = x* + 2x' 2 -4x-8, f{x)=x 3 + 3x 2 -4>, x=-2. 

 1 + 2 _4 -8(-2 1 + 3 +0 -4(-2 



-2 -0 









-2 -2 



—4 









1-2 



-2 4 









-2 2 



-2 









-1 



-2 









-2 



-4 









-3 





. *V- 



-2) 



4 







" u- 



-2)" 



"3 





"When the general symbol x is under the radical sign, the usual 

 method is to substitute a-\-h for x, and then to develope the 

 terms containing a + h by the binomial theorem, a being the 



value of x for which the fraction becomes ^-. But it will fre- 

 quently be more convenient and simple to proceed as in the fol- 

 lowing example. 



If —j-r — ttK 77T, which becomes - when x = a, be mul- 



V(a + 2x) — VZa? 



ti * M b y ^(l a +4+V3a (whi ^ f ° r X=a > ' l *\/l ] > the re " 



suiting fraction will be — = - ; hence, multiplying this by 



to "I io£\X — a) & 



\f 7y> we nave o\/ o ^ 0r * ne Y3 ^ ne °f the original fraction, 



when x=a. 



In like manner the fraction ^—^ may be rational- 



x — 1 J 



ized by multiplying it by ^(5^—1) + V% and j/(5x—1)+2; 



