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Fractiones algebraicae et fractiones partiales

Svmmarivm

Haec lectio calcvlvm fractionvm nvmericarvm ad qvotientes polynomiorvm extendit. Fractiones algebraicas earvmqve dominivm definit, demonstrat simplificationem factoris commvnis aeqvalitatem esse solvm in dominio originali, qvod pvncta svblata delineationvm earvm explicat, et aeqvalitatem dvarvm fractionvm per principivm identitatis describit. Svmmam cvm minimo commvni mvltiplo denominatorvm, prodvctvm, qvotientem fractionesqve compositas tractat, et aeqvationes cvm fractionibvs et cvm radicalibvs solvit, attendens ad solvtiones qvae orivntvr cvm denominatores tollvntvr avt ad qvadratvm elevatvr. Postremo demonstrat fractionem propriam cvivs denominator radices simplices habet vno tantvm modo in fractiones partiales decomponi, coefficientibvs qvi factore tegendo compvtantvr, decompositionem ad factores iteratos extendit, cvm propria vnicitatis eivs demonstratione, et ad factores qvadraticos irredvcibiles, eamqve ad svmmas telescopicas compvtandas adhibet.

Proposita discendi

  1. Dominivm fractionis algebraicae determinare, eam recte simplificare et pvncta svblata delineationis eivs agnoscere.
  2. Fractiones algebraicas addere, mvltiplicare et dividere fractionesqve compositas redvcere, adhibito minimo commvni mvltiplo denominatorvm.
  3. Aeqvationes cvm fractionibvs et cvm radicalibvs solvere, reiectis valoribvs exclvsis et solvtionibvs qvae elevatione ad qvadratvm additae svnt.
  4. Fractionem in fractiones partiales decomponere, decompositionem ivstificare eamqve ad svmmas telescopicas adhibere.

Fractiones algebraicae

Fractio algebraica est qvotiens dvorvm polynomiorvm, PQ\frac{P}{Q}, polynomio QQ a nihilo differente. Fvnctionem rationalem definit qvae cviqve nvmero reali xx valorem P(x)Q(x)\frac{P(x)}{Q(x)} attribvit, eivs generis qvod in lectione de fvnctionibvs algebraicis tractatvm est, et eivs dominivm constat ex nvmeris realibvs qvi denominatorem non evanescere facivnt; ceteri valores exclvsi vocantvr. Cvm polynomivm a nihilo differens gradvs nn radices habeat non plvres qvam nn, secvndvm lectionem de algebra polynomiorvm, valores exclvsi semper nvmero finiti svnt, et ad eos inveniendos denominator in factores resolvitvr artibvs lectionis de radicibvs polynomiorvm gradvs svperioris. In seqventibvs aeqvalitas inter fractiones algebraicas intellegitvr in omnibvs nvmeris realibvs in qvibvs ambo eivs membra definita svnt, et cvm dominia amborvm membrorvm non congrvvnt, id expresse indicatvr.

Simplificatio

Fractiones nvmericae simplificantvr nvmeratore et denominatore per factorem commvnem divisis, vt in lectione de nvmeris rationalibvs visvm est. In fractionibvs algebraicis eodem modo proceditvr, sed factor commvnis nvnc polynomivm est, qvod in qvibvsdam pvnctis evanescere potest; ex qvo seqvitvr fractionem simplificatam definitam esse posse vbi originalis non est.

LemmaSimplificatio

Sint PP, QQ et CC polynomia, QQ et CC a nihilo differentibvs. In omni nvmero reali qvi QCQC non evanescere facit, qvotiens PCQC\frac{PC}{QC} evndem valorem accipit ac PQ\frac{P}{Q}. Dominivm fractionis PCQC\frac{PC}{QC} est dominivm fractionis PQ\frac{P}{Q} sine radicibvs polynomii CC; itaqve ambae fractiones eadem fvnctio svnt solvm si secvnda ad dominivm primae restringitvr.

PCQC=PQ,Q(x)≠0∧C(x)≠0\frac{PC}{QC} = \frac{P}{Q}, \qquad Q(x) \neq 0 \wedge C(x) \neq 0

Adhibentvr lemma de signo prodvcti, ex lectione de fvnctionibvs algebraicis, et lemma amplificationis, ex lectione de nvmeris rationalibvs, qvod pariter de nvmeris realibvs valet. Figvra demonstrationem seqvitvr cvm P=x−3P = x - 3, Q=x−2Q = x - 2 et C=x+3C = x + 3.

Demonstratio

  1. Q(x0)C(x0)≠0↔Q(x0)≠0∧C(x0)≠0\dato{d}{Q(x_0) C(x_0) \neq 0 \leftrightarrow \resaltar{Q(x_0) \neq 0 \wedge C(x_0) \neq 0}}

    Sit x0x_0 nvmervs realis qvi QCQC non evanescere facit. Prodvctvm nvmerorvm realivm nvllvm est si et solvm si aliqvis eivs factor nvllvs est; vnde neqve QQ neqve CC in x0x_0 evanescit.

  2. P(x0) cQ(x0) c=P(x0)Q(x0),c=C(x0)≠0\frac{P(x_0) \, \resaltar{c}}{Q(x_0) \, \resaltar{c}} = \frac{P(x_0)}{Q(x_0)}, \qquad c = C(x_0) \neq 0

    Nvmervs c=C(x0)c = C(x_0) nvllvs non est, et ex lemmate amplificationis ambos terminos fractionis per evm mvltiplicare valorem eivs non mvtat.

  3. Dom⁡(PCQC)={x∈Dom⁡(PQ):C(x)≠0}\operatorname{Dom}\left( \frac{PC}{QC} \right) = \left\{ x \in \operatorname{Dom}\left( \frac{P}{Q} \right) : \resaltar{C(x) \neq 0} \right\}

    Eadem aeqvivalentia dominivm describit: nvmervs realis in dominio fractionis PCQC\frac{PC}{QC} est prorsvs cvm in dominio fractionis PQ\frac{P}{Q} est et CC non evanescere facit.

  4. C(r)=0∧Q(r)≠0→r∈Dom⁡(PQ)∧r∉Dom⁡(PCQC)C(r) = 0 \wedge Q(r) \neq 0 \rightarrow r \in \operatorname{Dom}\left( \frac{P}{Q} \right) \wedge \resaltar{r \notin \operatorname{Dom}\left( \frac{PC}{QC} \right)}

    In radice rr polynomii CC qvae radix polynomii QQ non est, fractio simplificata definita est et originalis non: delineatio originalis est delineatio simplificatae illo pvncto svblato, qvod vt circvlvs vacvvs pingitvr. In figvra, r=−3r = -3.

Qvid fvnctio originalis prope pvnctvm svblatvm faciat in lectione de tabvlis signorvm et inaeqvationibvs tractabitvr.

Solvm factores commvnes simplificantvr, et ideo simplificatio postvlat vt privs nvmerator et denominator in factores resolvantvr: addendvm commvne factor non est, et evm delere errorem parit. Ita, si littera xx in x+2x+4\frac{x + 2}{x + 4} deleretvr, valor constans 24\frac{2}{4} prodiret, cvm fractio valeat 23\frac{2}{3} in x=2x = 2 et 35\frac{3}{5} in x=1x = 1.

ExemplvmSimplificatio eivsqve pvnctvm svblatvm

Fractionem x2−9x2+x−6\frac{x^2 - 9}{x^2 + x - 6} simplificare et delineationem eivs describere. Figvra in recta valores exclvsos notat ac deinde fractionem simplificatam pingit cvm pvncto qvod originali deest.

Demonstratio

  1. x2−9x2+x−6=(x−3)(x+3)(x+3)(x−2)\frac{\resaltar{x^2 - 9}}{\resaltar{x^2 + x - 6}} = \frac{\resaltar{(x - 3)(x + 3)}}{\resaltar{(x + 3)(x - 2)}}

    Nvmeratorem, differentiam qvadratorvm, et denominatorem, trinomivm radicvm −3-3 et 22, in factores resolvimvs.

  2. (x+3)(x−2)≠0↔x≠−3∧x≠2\resaltar{(x + 3)(x - 2)} \neq 0 \leftrightarrow \dato{d}{\resaltar{x \neq -3 \wedge x \neq 2}}

    Valores exclvsi svnt radices denominatoris; dominivm est recta sine illis dvobvs pvnctis.

  3. x2−9x2+x−6=x−3x−2,x≠−3\frac{x^2 - 9}{x^2 + x - 6} = \resaltar{\frac{x - 3}{x - 2}}, \qquad \resaltar{x \neq -3}

    Ex lemmate simplificationis factor x+3x + 3, qvi in dominio non evanescit, tolli potest. Fractio simplificata maivs dominivm habet: solvm nvmervm 22 exclvdit.

  4. −3−3−3−2=−6−5=65\frac{\resaltar{-3} - 3}{\resaltar{-3} - 2} = \frac{-6}{-5} = \resaltar{\frac{6}{5}}

    In −3-3 fractio simplificata valet 65\frac{6}{5} et originalis definita non est: delineatio originalis est cvrva simplificatae sine pvncto abscissae −3-3.

Aeqvalitas dvarvm fractionvm

Dvae fractiones nvmericae pq\frac{p}{q} et rs\frac{r}{s} aeqvales svnt si et solvm si ps=qrps = qr. De fractionibvs algebraicis simile criterivm valet, cvivs demonstratio in facto qvodam de polynomiis nititvr: identitas qvae in omnibvs nvmeris realibvs praeter nvmervm finitvm eorvm valet in his qvoqve valet.

LemmaIdentitates praeter nvmervm finitvm pvnctorvm

Si dvo polynomia evndem valorem accipivnt in omnibvs nvmeris realibvs praeter, ad svmmvm, nvmervm finitvm eorvm, eosdem coefficientes habent et in omni nvmero reali evndem valorem accipivnt.

Per redvctionem ad absvrdvm ratiocinamvr cvm lemmate de radicibvs diversis ex lectione de algebra polynomiorvm. Figvra differentiam dvorvm polynomiorvm svpra axem xx pingit, cvm dvobvs pvnctis exceptis.

Demonstratio

  1. x∉{s1,…,sm}→D(x)=0\dato{d}{x \notin \{ s_1, \ldots, s_m \} \rightarrow \resaltar{D(x) = 0}}

    Sint AA et BB dvo polynomia, D=A−BD = A - B eorvm differentia et s1,…,sms_1, \ldots, s_m nvmeri reales in qvibvs fortasse non congrvvnt. Differentia in omnibvs ceteris evanescit.

  2. t1,…,tk+1∉{s1,…,sm}→D(t1)=⋯=D(tk+1)=0t_1, \ldots, t_{k + 1} \notin \{ s_1, \ldots, s_m \} \rightarrow \resaltar{D(t_1) = \cdots = D(t_{k + 1}) = 0}

    Si DD polynomivm nvllvm non esset, gradvm kk haberet et, ex lemmate de radicibvs diversis, radices non plvres qvam kk. Sed inter k+m+1k + m + 1 nvmeros natvrales 1,2,…,k+m+11, 2, \ldots, k + m + 1 svnt saltem k+1k + 1, qvos t1,…,tk+1t_1, \ldots, t_{k + 1} vocamvs, qvi excepti non svnt, et omnes radices polynomii DD svnt: qvod repvgnat.

  3. D=0→A(x)=B(x),x∈R\resaltar{D = 0} \rightarrow A(x) = B(x), \quad x \in \mathbb{R}

    Itaqve DD est polynomivm nvllvm: omnes eivs coefficientes nvlli svnt, et AA et BB coefficiens per coefficientem congrvvnt, vt in principio identitatis, et in omni nvmero reali evndem valorem accipivnt, etiam in s1,…,sms_1, \ldots, s_m.

TheoremaAeqvalitas fractionvm algebraicarvm

Sint PQ\frac{P}{Q} et RS\frac{R}{S} dvae fractiones algebraicae. Evndem valorem accipivnt in omni nvmero reali qvi QSQS non evanescere facit si et solvm si polynomia PSPS et QRQR aeqvalia svnt.

PQ=RS↔PS=QR\frac{P}{Q} = \frac{R}{S} \leftrightarrow PS = QR

Figvra demonstrationem seqvitvr cvm fractionibvs x2−9x2+x−6\frac{x^2 - 9}{x^2 + x - 6} et x−3x−2\frac{x - 3}{x - 2} exempli svperioris.

Demonstratio

  1. Q(x)S(x)≠0→P(x)S(x)=Q(x)R(x)\dato{m}{Q(x) S(x) \neq 0 \rightarrow \resaltar{P(x) S(x) = Q(x) R(x)}}

    Ponamvs dvas fractiones extra radices polynomii QSQS congrvere. Mvltiplicando per Q(x)S(x)Q(x) S(x) obtinetvr aeqvalitas inter valores dvorvm polynomiorvm, valida in omnibvs nvmeris realibvs praeter nvmervm finitvm, qvia QSQS nvllvm non est et nvmervm finitvm radicvm habet.

  2. PS=QR\resaltar{PS = QR}

    Ex lemmate de identitatibvs praeter nvmervm finitvm pvnctorvm, PSPS et QRQR idem polynomivm svnt. In exemplo ambo prodvcta svnt x3−2x2−9x+18x^3 - 2x^2 - 9x + 18.

  3. PS=QR∧Q(x)S(x)≠0→P(x)Q(x)=R(x)S(x)PS = QR \wedge Q(x) S(x) \neq 0 \rightarrow \resaltar{\frac{P(x)}{Q(x)} = \frac{R(x)}{S(x)}}

    Vicissim, si PS=QRPS = QR, satis est per Q(x)S(x)Q(x) S(x) dividere in nvmeris realibvs qvi id non evanescere facivnt, ex lemmate amplificationis.

Theorema ivstificat vt simplificatio decvssatim mvltiplicando confirmetvr. Lemma avtem clavis est decompositionis in fractiones partiales: permittit vt identitas in radicibvs denominatoris aestimetvr, vbi fractiones vnde oritvr definitae non svnt.