RATIONAL FRACTIONS. 



whence 



B + BM A = 1 

 2A-B = -5 

 A - B' = o 

 B = 3, B' = - I, and A = - I, 



therefore 

 confequeiitly 



1 -5 « _3 I L_ 



(l+x)'(l-.v) 1 + x' 1+* i-» 

 We have in both the preceding cafes extended the cal- 

 culation to a greater length than it is neceifary to obferve 

 in pradical cafes, in order that the reader might fee diU 

 tinaiy the principles on which the decompofition depends. 

 We fliall now furnifli him with an eafier pradical method, 

 which we haveextraaed from Bonnycaftle's Algebra, vol. i. 

 I. When the faaors of the denommator of the given frac- 

 tion are all unequal, or of the form 



N A A' A" . A'" 



^__ il_ _11_ +-^ t-_ll- 



D X — r' X — r' x — r" x — 



1+ &c. 



2. 



B' 



x-p 

 = -^, taking x in Q, and S = p. 



Let Q' = 



Q-FS 

 x-p 



, then 



B" = H-j taking * in Q', and S = /. 



Let Q" = 



Q' - B" S 



then 



^=-S = ^rTl?= I i being. = ^ 



N I + x' , . 



A" = -„- = , = - I ; being i = - x. 



The required fraaions are therefore 



— + 



I + *^ 



1 — X 1 + X X — x' 

 Example 2. — It is required to convert the rational frac- 

 tion — = r into its equivalent fimple 



D x'(i-x)'(i +x) ^ ^ 



fraaions. Here 



D 



ABB' B" 



+ - + —+-— + 



rx + 



I + X ' x^ ' x' ' X ' (l — xY ' 1 — x' 

 whence for i + x, the firft. faftor, we have 



A = — = , = — > * bemg = — i 



S x'-2x* + x' ±' ^ 



take that which conftitutes the denominator of the fimple 

 fraaion which is to be found, and let S denote the pro- 

 dua of all the remaining fraaions ; then if the root or 

 value of X, in that faaor, be fubftituted for .v in the 



formula— , it will give the numerator of the fraaion re- 



O 



2. If fome of the faaors are equal and others unequal, 

 or of the form ^ 



let S denote the produa of all the faaors m the deno- 

 minator, except one, as before ; then find the fimple fi-ac- 

 tions due to the unequal fraaions as above, and for thofe 

 of the equal faaors proceed as follows : 



N 

 I, B = -g-j taking x in N, and S =/. 

 o 



N-BS ^ 

 Let Q = -> then 



g ^ N ^ ^_ 



S I— X — x' + x 



Let now Q = 

 B'=§ = 



- = I, jc being = o 

 N-BS x + x^-x' 



x+f 



I + X - x" 



1 + X — n' 



S I — X — x^ + X 



Q - B' S 2x - x' 



Let Q' = ^ = 



^ X —p X 



O' 2- x' 



B" = ^ = 



S 1 —x~ x' + x' 



= I , X being = o. 



= 2, X being = o. 



Again, for (i — x)^ 



^^ S x' + x 



' - =— , x= being = i. 



Let now R = 

 + «' + * «' 



N-CS 



_ I ^' 



x'-^x 



i^» 



X — I 



= J + X 



C> = ^= , . 7 = h ■■* bemg = I. 



Therefore 



x' -f X* 



I _ I ' ^ ] 



x' (I - x)' (l + x) ~ "? ■*" ^ "^ X "*" 



-, as required. 



O" 



4. B'" = ^'. taking x in Q", and S = /. 



O 



&c. &c. 



Which operation being performed, the fum of the frac- 

 tions thus obtained, together with the former, will give all 

 the fimple fraaions into which the given fraaion 13 re- 

 folvable. ■ , r 



Example I.— It is required to convert the rational frac- 



I + x^ 

 3J 



tjon '■ into its equivalent fimple fraaions 



X — x' 



I + XI 



Here 



X — x' 



A A' A" 

 L ^ — } whence, by the 



o + X I — X I 4- X 



preceding rule 



^ ~ S" ""i-x" 



I ; being o = x 



2(1 -x)' "^ 4(i-x) 3(o+x) 



It is obvious, as we have before ftated, that this de- 

 compofition of rational fraaions mud neceffarily be afFeaed 

 as to its general application, by the imperfeaion of the 

 theory of equations, which will not admit of a praaical 

 refolution of the denominator into its fimple faaors in all 

 cafes. It will alfo further appear, that, in cafe of any of 

 the roots being imaginary, the fame will necefl'arily enter 

 into the numerators of the fimple fraaions. But this dif- 

 ficulty may be avoided : for fince the imaginary roots of 

 equations always enter in pairs, and the produa of fuch 

 pairs of roots being always real, being of the form 



x'- -+ 2 a .V + a'' 4- /3', 

 we may, inftead of refolving the fraaions into the fimple 



A A' A" 



fraaions -— -, -,, ;;. &c. refolve it into as many 



x — r x—r' X — r" 

 of thefe fraaions as it has rational fimple faaors, and into 



B .f + C . . 



as many fraaions of the form — — — — j-j-^, as it has 



pairs of iraagmary fadors, and then proceed as before. 



