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Aeqvationes polynomiales complexae: formvlae Cardani et Ferrarii

Svmmarivm

Haec lectio aeqvationes polynomiales in corpore nvmerorvm complexorvm tractat. Ex theoremate fvndamentali algebrae, qvod admittitvr, per indvctionem dedvcit omne polynomivm gradvs nn prodvctvm esse nn factorvm linearivm et exacte nn radices habere cvm mvltiplicitate sva nvmeratas; relationes Vietae pro qvovis gradv demonstrat, nec non formvlas svmmarvm qvadratorvm et cvborvm radicvm. Aeqvationes binomias, trinomias et reciprocas solvit eas ad alias gradvs minoris redigendo. Postremo aeqvationem tertii gradvs ad formam depressam redigit, formvlam Cardani dedvcit, radices per discriminans distingvit, aeqvatione Bombellii ostendit cvr formvla nvmeros complexos accipere coegerit etiam cvm omnes radices reales sint, et hvnc casvm formvla angvli tripli solvit. Deniqve aeqvationem qvarti gradvs ad formam depressam redigit, methodvm Ferrarii demonstrat, qvae eam in dvas aeqvationes secvndi gradvs per radicem resolventis cvbicae eivs dissolvit, et explicat cvr a gradv qvinto nvlla similis formvla exstet.

Proposita discendi

  1. Ex theoremate fvndamentali algebrae factorizationem omnis polynomii complexi in factores lineares dedvcere, et radices cvm coefficientibvs per relationes Vietae et svmmas potentiarvm conivngere.
  2. Aeqvationes binomias, trinomias et reciprocas solvere eas ad aeqvationes gradvs minoris redigendo.
  3. Aeqvationem tertii gradvs ad formam depressam redigere, formvlam Cardani dedvcere et radices cvbicae realis secvndvm signvm discriminantis eivs distingvere.
  4. Cvbicam tribvs radicibvs realibvs praeditam methodo trigonometrica solvere, et exemplo Bombellii explicare qvod mvnvs historicvm formvla Cardani in nvmeris complexis accipiendis gesserit.
  5. Aeqvationes qvarti gradvs methodo Ferrarii solvere, eas ad formam depressam redigendo et per radicem resolventis cvbicae in factores resolvendo.

Theorema fvndamentale algebrae in complexis

In lectione cvrsvs Algebra et Geometria I de radicibvs polynomiorvm gradvs svperioris theorema fvndamentale algebrae envntiatvm est, advmbrata est eivs conseqventia omne polynomivm complexvm gradvs nn prodvctvm esse nn factorvm linearivm, et ex eo dedvctvm est omne polynomivm reale prodvctvm esse factorvm linearivm et qvadraticorvm irredvcibilivm; hic demonstratio completa, per indvctionem, illivs factorizationis in complexis traditvr. Haec lectio rationem complexorvm seqvitvr: coefficientes et radices complexi esse possvnt, et qvaestio non iam est vtrvm aeqvatio solvtionem habeat, sed qvomodo omnes eivs solvtiones inveniantvr. Polynomivm coefficientibvs complexis gradvs nn est expressio P(z)=anzn+⋯+a1z+a0P(z) = a_n z^n + \cdots + a_1 z + a_0, cvm aka_k complexis et an≠0a_n \neq 0; nvmervs complexvs z0z_0 est radix ipsivs PP cvm P(z0)=0P(z_0) = 0.

Demonstrationes algorithmi divisionis, theorematis residvi et theorematis factoris, in lectione cvrsvs I de algebra polynomiorvm datae, solis proprietatibvs svmmae, prodvcti et divisionis coefficientivm vtvntvr, qvas C\mathbb{C} cvm R\mathbb{R} commvnes habet secvndvm lectionem de nvmeris complexis. Itaqve sine vlla mvtatione valent pro polynomiis coefficientibvs complexis, et idem accidit lemmati de radicibvs diversis et principio identitatis, eivsdem lectionis (dvo polynomia qvae in omni complexo evndem valorem capivnt eosdem coefficientes habent), et definitioni mvltiplicitatis radicis, ex lectione de radicibvs polynomiorvm. Qvod C\mathbb{C} distingvit est theorema infra scriptvm.

Factvm admissvm (theorema fvndamentale algebrae). Omne polynomivm gradvs n≥1n \geq 1 coefficientibvs complexis saltem vnam radicem complexam habet.

Vt in lectione cvrsvs I de radicibvs polynomiorvm, hoc theorema sine demonstratione admittitvr: omnes eivs demonstrationes notae instrvmentis vtvntvr qvae hvnc cvrsvm excedvnt, vt continvitate fvnctionvm dvarvm variabilivm realivm avt stvdio fvnctionvm variabilis complexae. Carolvs Fridericvs Gavss anno 1799 primam demonstrationem fere completam dedit, post conatvs Alembertii et Evleri. Praecipva eivs conseqventia est aeqvationes polynomiales non cogere vt C\mathbb{C} amplificetvr, sicvt x2=2x^2 = 2 coegit vt Q\mathbb{Q} amplificaretvr et x2=−1x^2 = -1 vt R\mathbb{R}: omne polynomivm complexvm in C\mathbb{C} penitvs in factores dissolvitvr.

TheoremaFactorizatio in complexis

Omne polynomivm PP gradvs n≥1n \geq 1 coefficientibvs complexis et coefficiente principali ana_n est prodvctvm ipsivs ana_n et nn factorvm linearivm, vt in formvla infra scripta, cvm z1,z2,…,znz_1, z_2, \ldots, z_n complexis non necessario diversis. Radices ipsivs PP svnt exacte nvmeri zkz_k, et qvaeqve in indice toties occvrrit qvoties eivs mvltiplicitas indicat; itaqve PP exacte nn radices complexas habet cvm mvltiplicitate sva nvmeratas.

P(z)=an(z−z1)(z−z2)⋯(z−zn)P(z) = a_n (z - z_1)(z - z_2) \cdots (z - z_n)

Demonstratio per indvctionem in gradvm procedit, vt in lectione cvrsvs I de svmmationibvs, progressionibvs et indvctione. Figvra eam in plano complexo comitatvr cvm polynomio P(z)=z5+2z4+2z3−4z2−11z+10P(z) = z^5 + 2z^4 + 2z^3 - 4z^2 - 11z + 10, cvivs radices svnt 11, qvae dvplex est, −2-2, −1+2i-1 + 2i et −1−2i-1 - 2i.

Demonstratio

  1. a1z+a0=a1(z+a0a1)\resaltar{a_1 z + a_0} = \resaltar{a_1 \left( z + \frac{a_0}{a_1} \right)}

    Casvs initialis: polynomivm gradvs 11 est coefficiens eivs principalis in vnicvm factorem linearem dvctvs, cvivs radix est −a0a1-\frac{a_0}{a_1}.

  2. P(z1)=0→P(z)=(z−z1)C(z),P(z)=(z−1)(z4+3z3+5z2+z−10)P(z_1) = 0 \rightarrow P(z) = \resaltar{(z - z_1)} \dato{c}{\resaltar{C(z)}}, \qquad P(z) = (z - 1)\left( z^4 + 3z^3 + 5z^2 + z - 10 \right)

    Ponamvs theorema vervm esse pro polynomiis gradvs nn, et sit PP gradvs n+1n + 1. Per theorema fvndamentale algebrae, PP radicem z1z_1 habet; per theorema factoris, PP est prodvctvm ipsivs z−z1z - z_1 et qvotientis CC gradvs nn, cvivs coefficiens principalis idem est ac ipsivs PP, vt terminis gradvs maximi comparatis apparet. In exemplo, z1=1z_1 = 1, et divisio synthetica qvotientem formvlae dat.

  3. P(z)=(z−z1)an+1(z−z2)⋯(z−zn+1)=an+1(z−z1)(z−z2)⋯(z−zn+1)P(z) = (z - z_1) \resaltar{a_{n + 1} (z - z_2) \cdots (z - z_{n + 1})} = \dato{f}{\resaltar{a_{n + 1} (z - z_1)(z - z_2) \cdots (z - z_{n + 1})}}

    Per hypothesim indvctionis, CC est coefficiens eivs principalis an+1a_{n + 1} in nn factores lineares dvctvs; svbstitvtione facta, PP scribitvr vt volebamvs. Per principivm indvctionis, factorizatio pro omni gradv n≥1n \geq 1 exstat.

  4. P(w)=an(w−z1)⋯(w−zn)=0↔w=z1∨⋯∨w=znP(w) = a_n (w - z_1) \cdots (w - z_n) = 0 \leftrightarrow \resaltar{w = z_1 \vee \cdots \vee w = z_n}

    Prodvctvm complexorvm nvllvm est tantvm si aliqvis eivs factor nvllvs est, et an≠0a_n \neq 0: nvmervs ww est radix ipsivs PP si et solvm si cvm aliqvo ex zkz_k congrvit. Nvllae aliae radices svnt praeter eas indicis.

  5. P(z)=an(z−r1)k1⋯(z−rs)ks,k1+k2+⋯+ks=nP(z) = a_n \resaltar{(z - r_1)^{k_1}} \cdots (z - r_s)^{k_s}, \qquad \resaltar{k_1 + k_2 + \cdots + k_s = n}

    Factores aeqvales conivngimvs: si r1,…,rsr_1, \ldots, r_s svnt radices diversae et rjr_j occvrrit kjk_j vicibvs, qvotiens ipsivs PP per (z−r1)k1(z - r_1)^{k_1} in r1r_1 non evanescit, qvia prodvctvm est factorvm non nvllorvm; itaqve k1k_1 est mvltiplicitas ipsivs r1r_1, et idem de ceteris valet. Mvltiplicitates svmmam nn habent; in exemplo, 2+1+1+1=52 + 1 + 1 + 1 = 5.

Cvm mvltiplicitas cvivsqve radicis bene definita sit, secvndvm lectionem cvrsvs I de radicibvs polynomiorvm, index z1,…,znz_1, \ldots, z_n ab PP determinatvr, ordine excepto: factorizatio vnica est. Posthac radices polynomii gradvs nn hvnc indicem nn nvmerorvm significant, cvm repetitionibvs qvas mvltiplicitas indicat. Si coefficientes reales svnt, radices non reales in paribvs conivgatis occvrrvnt, per theorema radicvm conivgatarvm lectionis de nvmeris complexis; dvobvs factoribvs cvivsqve paris inter se mvltiplicatis factor qvadraticvs realis obtinetvr, et sic factorizatio in realibvs lectionis de radicibvs polynomiorvm recvperatvr.