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Algebra polynomiorvm realivm

Svmmarivm

Haec lectio polynomia coefficientivm realivm tractat: eorvm definitionem per coefficientes, gradvm et aeqvalitatem inter ea; operationes svmmae, differentiae et prodvcti, cvm regvla qvae gradvm prodvcti dat et cvm earvm consensv cvm valoribvs nvmericis; algorithmvm divisionis, cvm theoremate residvi et theoremate factoris; nvmervm radicvm et principivm identitatis, qvod sinit vnvmqvodqve polynomivm cvm fvnctione qvam definit identificari et ostendit, exempli gratia, radicem qvadratam fvnctionem rationalem non esse; et factorizationem, cvm prodvctis notabilibvs. Ostenditvr praeterea polynomivm in factores resolvi posse etiamsi radices reales non habeat.

Proposita discendi

  1. Gradvm, coefficientes et coefficientem principalem polynomii agnoscere, et discernere qvando dvo polynomia aeqvalia sint.
  2. Polynomia addere, svbtrahere et mvltiplicare, et gradvm eventvs praedicere.
  3. Polynomia per algorithmvm divisionis dividere et theorema residvi et theorema factoris adhibere.
  4. Demonstrare polynomivm non nvllvm gradvs nn non plvres qvam nn radices habere, et principio identitatis vti ad coefficientes comparandos.
  5. Polynomia in factores resolvere per eorvm radices et per prodvcta notabilia.
  6. Qvaestiones applicatas per polynomia constitvere, vt volvmen capsae vel pecvniam vsvra composita avctam, easqve per theorema residvi et theorema factoris solvere.

Polynomia

Polynomivm coefficientivm realivm in variabili xx est expressio formae seqventis, vbi nn est nvmervs natvralis vel nihil et coefficientes a0,a1,…,ana_0, a_1, \ldots, a_n svnt nvmeri reales:

P(x)=∑i=0naixi=a0+a1x+a2x2+⋯+anxnP(x) = \sum_{i = 0}^{n} a_i x^i = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n

Qvodqve addendvm aixia_i x^i terminvs vocatvr, et a0a_0 est terminvs constans. Qvodqve polynomivm fvnctionem definit: eam qvae cviqve reali xx nvmervm P(x)P(x) attribvit qvi oritvr cvm in expressione variabilis per xx svbstitvitvr. Fvnctiones sic obtentae, polynomiales appellatae, svnt fvnctiones algebraicae simplicissimae, in lectione de fvnctionibvs algebraicis propositae.

Dvo polynomia svnt aeqvalia cvm eosdem coefficientes habent, potentiam pro potentia, postqvam potentiae qvae in altervtro desvnt coefficientibvs nvllis completae svnt; ita 1+0x+3x21 + 0x + 3x^2 et 1+3x21 + 3x^2 idem polynomivm svnt. Polynomivm cvivs omnes coefficientes nihil svnt polynomivm nvllvm vocatvr.

Gradvs polynomii non nvlli est maximvs exponens cvivs coefficiens a nihilo differt; hic coefficiens coefficiens principalis vocatvr. Ita P(x)=3−x+5x4P(x) = 3 - x + 5x^4 gradvm 44 et coefficientem principalem 55 habet, et scribitvr deg⁡(P)=4\deg(P) = 4. Polynomivm nvllvm gradvm non habet, qvia nvllvm coefficientem a nihilo differentem habet. Cvm gradvs et coefficiens principalis in coefficientibvs legantvr, et dvo polynomia aeqvalia eosdem coefficientes habeant, vtrvmqve bene definitvm est.

Iam vero polynomivm fere semper per fvnctionem qvam definit adhibetvr, et qvaeri potest nvm dvo polynomia diversa eandem fvnctionem definire possint; qvod si ita esset, gradvs fvnctionis polynomialis non bene definitvs esset. Principivm identitatis, qvod in sectione de nvmero radicvm demonstratvr, respondet non posse: dvo polynomia eandem fvnctionem definivnt si et solvm si aeqvalia svnt. Vsqve ad id, qvidqvid de polynomiis dicitvr, et praesertim de eorvm gradv, ad coefficientes refertvr, ita vt nvlla demonstratio illo principio prior ex eo pendeat. Postea qvodqve polynomivm cvm sva fvnctione identificari potest, et ivre coefficientes in aeqvalitate inter fvnctiones polynomiales comparantvr, vt in problemate solvto 5.

Distribvtio

Polynomia primvm secvndvm gradvm distribvvntvr:

  1. Constantia: ea gradvs 00, formae P(x)=cP(x) = c cvm c≠0c \neq 0, vna cvm polynomio nvllo.
  2. Linearia, gradvs 11: P(x)=a0+a1xP(x) = a_0 + a_1 x, cvm a1≠0a_1 \neq 0.
  3. Qvadratica, gradvs 22, et cvbica, gradvs 33; ea gradvvm svperiorvm svo nvmero nominantvr: gradvs 44, gradvs 55, et sic deinceps.

Secvndvm nvmervm terminorvm non nvllorvm, polynomivm monomivm vocatvr si vnvm habet, binomivm si dvos, trinomivm si tres.