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NX, , q, , , , , PARTIAL FRACTION, , , , , 4 Introduction, P(x), , Rational Fraction : If P(x) and Q(x) are polynomials in x then Q(x), , jscalled a rational fractions. ., a text, x'+5x°+2x+1 «atl fect, e.g. Pe a + 2x43 are rational fractions., , proper Fraction : If degree of P(x) is less than degree of Q(x), then, , Fat is called a proper fraction., , oe fracti, &y + 5x +6 is a proper ction., , improper Fraction :, , | « Ifdegree of P(x) is greater or equal to nee of Q(x), then aa, , ealied an improper fraction., , x +2x+3 : 4, e.g. See is an improper fraction., , Fa, , yao

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4-2 Partial Fro,, Ty, , Basic Mathematics, , Improper fraction to Proper fraction : Any improper fraction cant, , expressed as sum of a polynomial and a proper fraction on Civ, method., , ; R, , ie. Improper fraction = Q+ D, , . , Remainder, = Quotient +” Divisor, , . P(x, Partial Fraction : Every proper fraction Qa) can be expressed ing, , sum of difference of its sub-proper fractions according to factors g, , Q(x). These sub-proper fractions are called partial fraction of. an}, , 1 -1 1, FHA &D &D, -1 1 s 1, Here G42) (+1) 8 Proper fraction of 65) (x + 1), , Case 1 : Non-Repeated Linear Factor in the, Denominator, , 42, , , , , If the denominator contains non-repeated linear factor of the typ, , i b) then for each such factor there is Partial fraction of the typ, , , , (ax+b), , In general., i) ———_Nn____A B Cc, ” GDR K+ EH EEO, , i) >A, (ax +b) (x +4) “(ax+b) *x+a, , where A, B, C etc, are constant to be determined,, , , , a, T, , Technical sone, Publications — An up thrust for knowledge, , 4-3, Partial Fraction, , eS, , example 4.1 :, , , , , Resolve into partial fractions ~—2_, x(x ~ 1), , 5 _MSBTE? Winter 2006, Summer 2040), , sii ee, Given <(x— 1) |S @ Proper fraction. Here factors of, , , , , , , golution +, , denominator are linear and unequal., X=2_A,_B, x(x- 1) x" (x=1) -@, 422 _ AG-1)+ BQ), xx-1) x(x 1), , Comparing numerators, x-2 = A(x-1)+B(x) .Q, , To find A, Put x = 0 in equation (2), we get, 0-2=A(-1)+B(0), -2=A(-1), -2=-A, = A=2, To find B, Put x = 1 in equation (2), we get, 1-2 = A()+B(1), -1=B, , , , B=-1, , Putting the value of A and B in equation (1), x20 22d, x(x-1) x (x-1), , , , f T, Example 4.2: Resolve into partial fractions 7, , , , — An uo thaust for knowledge

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aed Partic, , Is ., Solution : Given fraction [> “7 is 4 proper fraction., , 1 1, X-1) &-N&+)), _a’—b' = @+b) (@-b), , Here the factors of denominator are linear and unequal., , i A BT, , @&-D&+ &-1 w+) (, 1 _ A(e+1)+B(x- 1), , @-ND&t) &-D&t), , Comparing numerators, , , , , , , , , , Let, , 1=AQ-1)+Ba-1) Q), To find A, Put x = | in equation (2), we get, 1= AC +1)+BQ@), 1=AQ), 1, , 2, To find B, Put x =— 1 in equation (2), we get, 1=A(0)+B-1-1), -1=Be-2), , , , Putting the value of A and B in equation (1), , Fg A/D 12, x-1> (&x-1) «th, rap gies 1, x1 2K-1)7 2K+1), , ™, T Technical Publications” — ‘An up thrust for knowledge, , , , , , ; $420 All 52, solution: Given fraction x — x '84 proper fraction., , 1 1, =x x=, , Here the factors of denominator are linear and ‘unequal., 1 A B, , , , , , Tet x1) x*@-1) 20), 1 _ A(~1) +B), x(x— 1) x(x-1), Comparing numerators, 1 = A(x- 1) + B(x) Q), , To find A, Put x = 0 in equation (2), we get, 1 = A(O- 1) + B(0), 1=AC 1), A=-A, , a A=-1, , To find B, Put x = 1 in equation (2), we get, 1 = A(0) + BCI), 1=B()), B=l, , Putting the value of A and B in equation (1), Sedo” Slee, , , , zoe St), xex x @-1), , —— —=—— ——=—, LJ Technical Publications — Anup thrust for knowledge

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46, , , , Basic Mathematics, , Partial Fraction i patna. 427 Pert Fr, resolve into partial fractions +3, amp 4s: Re Pi ions a oy 3, fi, , solve into partial fractions wut, eg weer x Sairiner 2008, Winter 2012, , , , , Example 44: R, , , , Solution : Given fracion T7353 is a proper fraction., , Here the factors of aan are linear and unequal., , 1, ea are, 1 AN BL 0, Let eraRee wD wD *, _ x +1) + Bxt+2, aan" +2 «+1, , Comparing numerators, Z, 1= A+ 1) +BOt2) --- Q), , To find A, Put x = - 2 in equation (2), we get, 1=A(-2+1)+BO), 1=AC1), , 1=-A, x =-l, To find B, Put x =~ | in equation (2), we get, 1 = A(0)+ B(-1+2), 1 =B(1), ei B=1, Putting the value of A and B in equation (1), we get, , 1 = 1, P43x42 42) Ht), , , , Technical Publications - An up thrust for knowledge, , , , i To aRchso rt, Given fraction 57 —- 2x — 3 88 Proper fraction., , , , soution +, Here the factors of ‘denominator are linear and unequal., 2xt3 2x +3, x -2x-3 (K+) &-3), 2x+3 eof B, lt &FN«-3) =&+D*@-3) »), —2x+3 AG 3) + BOD), «+ Dar 3) (x+1)(«-3), Comparing numerators, svi), , 2x43 = A(x—3)+ Bet!), Tofind A, Putx =— 1 in equation (2), we get, A-1)+3= A(-1-3)+B@), , -2+3=AC4), 1=AC4), A=-14, , Tofind B, Put x = 3 in equation (2), we eet, 2(3) +3 = A@)+BG +1), 6+3 =B(4), 9 = B(A), B= 9/4, hating the value of A and B in equation (1),

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Partial, Basic Mathematics 4:8 eral, X+4xtt, , Example 4.6: Resolve into partial fractions & Next 1) = 4) x + 1) (+3), , , , sa (MSBTE ; Summer 2014, , x+4xt1, , Solution : Given fraction Neat Da&t3), , Here the factors of denominator are linear and unequal., , __xtde+t A BE, R-Deat Dats) &-D &+D (x +3), , . x+4x+1 _AG+DK+3)+ B= ND &FI+CR= DK, “RDetDET) (-N&+NR+3), Comparing numerators, x4 4xt1 = A(x +1) (x+3)+BX- 1) (+3), +C(x-1) (+1), , , , Let, , To find A, Put x = | in equation (2), we get, +41) +1 = A(I +1) (1 +3)+B(0) + C(O), 1+4+1=A(2)(4), 6 = A(8), A= 6/8, . A = 3/4, To find B, Put x = — ] in equation (2), we get, (-1P + 4-1) +1 = A() + BE 1-1) (1 +3) + C0), 1-441 = BE2)Q), -2=B-4), 2 = B(4), , 2, =4 « Be, , Nie, , , , +, , Technical Publicatic ‘, isl Publications — An up thrust for knowledge, , is a proper fraction., , ve (I), , ), , -- (2), , , , 4-9, Pa, , we, Put x =— 3 in equation (2), we get, , , , , , rofind C, 3) + 4(- 3) + 1 = AQ) + BOO) + C(- 3-1) +(-3 +1), 9-12+1=C(-4)(-2), -2=C(8), 22 -1, ie “8 6 Cae, puting the value of A, B and C in equation (1), xt+4xt 34 . 12 4, Reet DET) &=-DN*R+D HHH), x+4x+1 Qo FB, an, , , , cK eee, “@-D&+D&+3) 4-1) 2 +1) 4K +3), , 5 (MSBTE: Summer 1985, , -3, LT: into ial, Example 4.7: Resolve int partial fractions aay 1) (@x+3), , ‘ i . 2x3 EEE, Solution: Given fraction Gf 1) Qx+3) is a proper fraction., , Here the factors of denominator are linear and unequal., , 2x-3 2x-3, @-DO@x+3) &+)&-DExt3), ab? = (a+b) (a-d), , 2x -3 WAS cB go Ge, * Bo-DEAT™ @+D* (2x +3), , ee St x — 1) (2x +3)+Bxt DN 2x+3 + Ct, {+N &-1) @x3)~ (e+ D@-DEX+d), , 2x-3= Ag oer oe Bet Det, , , , , , , , , , «(BD, , , , ; +C(x+D&-D 2), find A, Put x =— 1 in equation (2), we get, 2-1) -3 = AE 1- DAE N+3) + BO) + CO), , Technical Publications” — An up thrust for knowiedge