Partial Fraction Decomposition.pdf

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Unformatted text preview: PARTML FKACTmM DEcomPosonN LQJC Qt“ ‘- _P(;o wherc Pun] Qua 0x1 po\7nomTq\3 .GJM SULM Wm‘\ (1)3“)th < 9&3 (81x3 . (Lt. *0“ is ex froeer‘ rcfix‘onmk “Fug We. wish *5 Kn+Q3¢aJTQ Pun ch: QCX) Prggedxum -' ‘ _ P ag'Vor (Db/q WW0 \rrecxm‘ficfig {ZLLLJMFS . \A/flJrL Hag POJJM‘Q\ «CrmL‘HOn gegomposfihn (£01‘m3 (5" P()c\/ QIXX - SO‘VC. {-or ‘HmL UMKhown LonficxvfiS §n+foduggcx TA ‘33an 2*. 1 vi ear QJVQ . Linear Factors Let R(x) = R(x)] Q(x)"be aproper rational fraction, and suppose that the linear factor ax + b occurs in times in the complete factorization of Q(x). That is, (ax + b)" is the highest p0wer of rat + ,b that divides “evenly” into Q(x). In this case we call n the multiplicity of the factor ax + b. RULE 1 Linear Factor Partial Fractions The part of the partial—fraction decomposition of R(x) that corresponds to the linear factor ax + b of multiplicity n is a sum of n partial fractions, specifically A1 + ————A2 + + —-——A" (6 ax + b (ax + b)2 («it +.b)"’ - ) where A, A2, . . . . A, are constants. If all the factors of Q(x) are linear, then the partial—fraction decomposition _ ' R(x) is a sum of expressions like the one in Eq. (6). The situationis especially sisap if each of these linear factors is nonrepeated—that is, if each has multiplicity n =. In this case, the expression in (6) reduces to its first term, and the partial-tract? _ decomposition of R(x) is a sum of such terms. Quadratic Factors Suppose that R(xh = Ptx)! th'} is a proper tationalfraction and that the hredusiblt Quadratic factor ax1+bx+c occursntimes in the factorization. That is. (cur2 +bx H)" is the highest power of run:2 + bx + c-that divides evenly into QUE). As before, wees]! n. the multiplicity of the quadratic factor axz + bx + c. RULE 2 Quadratic Factor Partial Fractions The part of the partial-fraction decomposition of R(x) that corresponds to the irreducible quadratic factor ax2 + bx + c of multiplicity n is a sum of n partial fractions. It has the form le +,C1 321 + C: Bax + C5 —T—_+_i'2”*——2+”'+-—_'—’ (8) ax +bx+c (ax +bx+c) . (ax2+bx+c)" where B1, 32, . . ., B,,, C1, C2. .. .. and C,I are constants. 1f Q(x) has. both linear and irreducible quadratic factors, then the partial- fraction decomposition of R(x) is simply the sum of the expressiOns of. the form in (6) that correspond to the linear factors plus the sum of the expressions of the ~ form in (8) that correSpond to the quadratic factors.» In the case of aniitreduciblc quadratic factor of multiplicity n = 1, the expression in (8:) reduces to its first term alone. Note was the above. (Jess jenet‘ollze, in maker degree ‘Y’Cxci‘cK-S - I. 61:) Gxtkhfl 1 Axilbx‘tcxd) + (xumVx—sx x“~»>< t \ (a (\C‘i‘x’}; Akhén ‘ , ~ " J: .. I . . . it ' [ff/UM! ARC, x Kgflihi <. airs/sax was“ iAhw we Nanci“: i") E X’3 4‘, c OWN“ ...
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