the fimplc fraaions foiiglit.' Let A, A', A", A'", &c. re 

 prclcnt their numerators, fo that 



N_ ^ A' A" . . A'"-" 



RATIONAL FRACTIONS. 



whence A = i, and A' — A = i, or A' = 2, therefore 



+ &c. 



fluent '-^ + flu. _ 4. flu. 



Reduce now tlicfe feveral fimple fraftions to a common 

 denominator by the ufual rule, and add their fovcral nu- 

 merators together ; then comparing the co-efficients of the 

 like powers of x in the numerator of this new fraction, with 

 thofe in the numerator propofed, we ihall obtain fufficient 

 equations for determining the proper values of A, A', A", 

 A'", &c. and the decompofition will be cffedted as pro- 

 pofed. This will be better undcrRood from a partial 

 example. 



Let us therefore afFume the fraftion '--: — ; , 



X' — Ox' + n X — 6 



which it is required to refolve into its fimple fraftions. 



Here the roots of the denominator, by making it equal 



to zero, are i, 2, and — 3, whence the propofed equation 



x' — 4.-C — 6 



(' 



0^ ,:|(«-I)X 



which are known. 



Required the fluent of , _. „. 



I — 4.x; H- 2*' i— 2x-fx' 



Here, by making x'^ — 2 x + 'j — o, we have x = l + 

 5 \/2 ; make, therefore. 



-, or ol f 



(i«-^)x 



A. 



+ 



A'; 



^ — 2x + x'' X — I ~ ^^2 ' X — I + i^^i 



- ^ {"-l + h ^2 )i + A' {x-l-\^z)i 

 i — 2X + x' 



Here we have A-fA'= — i(i — iy2)A + (I.f- 

 5 v'2) A' = — i, whence we have A = — J, and A' = 



( »^ I ) JC 



— ^, therefore fluent of ^-^ ~~ = flu. 



-ix 



may be put under the form 



Make now this fraftion 

 A' A" 



- — - -4- 



X — 2 X + 3 



(.V- 1) (x - 2) {x + S) 



— 4x — 6 A 



+ 



+ + 



I X — 2 K -f- 3 



4X-6). 



3^ 



ingr the fluents of -f -|- , , 



* K— I X — 2 X + ^ 



known, being hyp. log. (.v — i) + 2 hyp. log. (* — 2) -f- 



3hyp.log.(x -h 3) = hyp.log. [(j:- 1) (* - 2)'(x+ 3)'] 



The fame method may be employed for all rational 



fradions, whofe denominators arc refolvable into unequal 



fimple faftors, but as this cannot be generally efFefted when 



the higheft power of the variable quantity exceeds the 



fourth degree, we are neceffarily limited in our application 



of the rule, in confequence of the imperfedlion which ftill 



attends the general folution of equations. In the cafe, 



however, of equal faftors in the denominator, a different 



procefs is required, which we will explain, after illuftrating 



the above rule by two or three examples. 



Required jhe fluent of -— by refolving it into 



its fimple fraftions. 



Here the factors of the denominator being x and I — *, 

 we make 



{i + x)i Ai A'^ _(A-t- (A'-A)x)i 



• r-= + ■ — T J 



X — x' X 1 — X X — ir 



* — I + !•« 



+ 



~ki 



-, which are known. 



[x- i) {x- 2)[x + 3) x-l 

 which being reduced to 3 common 

 «' — 4 jr — 6 



denominator, and added together, give , , „ , 



_ A(a:- 2) {x + 3) 4 A'(.v- 1) (« + 3) -t- A"(x- 1) {x-2) 



~ ■ .x' — 6 «* -t- 1 1 K — 6 



= (A -1- A' + A") .v' + (A + 2 A' - 3 A") .V- (6 A 4- 

 3 A' — 2 A" ) whence 



A4- A'4- A"r:= 6 



A + 2 A' - 3 A" = - 4 

 6A -1- 3A'-2A''= 6 

 from which we readily draw A = I, A' ^ 2, A" = 3 } fo 



I 2 3 



that the required fraaions are ' ' 



(6x- ■ 



If therefore the fluent of -^^ — ?^H '—7^ were re- 



s:'— Ox-f Ilx — O 



quired, we might immediately reduce the problem to find- 



which are 



•■« — I — ^ v/2 



We have hitherto confidered the faftors of the denomi- 

 nators of the propofed fraftions to be all unequal; when any 

 number of them are equal, a little difference in the oper. 

 ation is then required ; for it is obvious that if two of our 



A A' 



fraftions were of the fame form, for example -\ , 



X — r X — r 



A -1- A' 



thefe two would form, in faft, but one fraftion , 



X — r 



which leads us to no ufeful refult. 



In cafe of equal faftors, therefore, we muft proceed i» 



follows : let the propofed fraftion be 



a-i-ix+cx'- + dx'' + yx"- ' 



(7^)»(.v-r)(x-r')(x-r") *^- 

 A Hume it equal to 

 B B' B" Bf"-'» 



orr7)„ + (^^)" - . .+ (x-py-' + ■ 



A A' A" 



-P 



+ 



j + 



-„ + &c. 



X — r X = r X 



Where the upper line reprefents the fraftions due to the 



« equal faftors, and the lower thofe due to the — unequal 

 ^ n 



faftors. 



Let now thefe fraftions be reduced to a common deno- 



minator, and the co-efficients equated as before, which will 



give us the required values of the feveral numerator* 



fought. 



1 — Ca: 

 Example. — It is required to convert — 



into its equivalent fimple fraftions. 

 AfTume ^ 



(I -x) (I -!-«)' 

 B' A 



\l+xY{l-x) ~{I + xf + (l+~x) ■•" (I-X) 



- E + (' + *) B' A _ 



~ (i +x)^ "^ I-X ~ 



B(i- x)-h B'(r-x^)4- A(i4- 2x + x') _ 



[i+xy{x-x) 

 B -I- B' -H A -f (2 A - B}x -KA - BQ x" 



(n-x)Mi-*) 



3 N 2 whence 



