Fvnctiones algebraicae et polynomia: synthesis et provocationes

Svmmarivm

Haec lectio partem de fvnctionibvs algebraicis et polynomiis conclvdit. In vnam catenam dedvctivam colligit qvae in novem eivs lectionibvs demonstrata svnt, a notione fvnctionis et ab algorithmo divisionis vsqve ad integram tractationem fvnctionvm rationalivm et fvnctionvm cvm radicalibvs; ostendit qvomodo pars nvmeris natvralibvs et generibvs nvmerorvm partivm priorvm nitatvr, sincere indicat effectvs qvi sine demonstratione admissi svnt et explicat qvomodo cvm geometria analytica plani conectatvr. Finitvr decem provocationibvs, difficilioribvs qvam problemata proposita partis, qvae postvlant vt plvres lectiones conivngantvr et idea inveniatvr.

Proposita discendi

Catenam definitionvm et theorematvm partis restitvere, explicando qvis effectvs cvi nitatvr.
Theoremata de polynomiis cvm analogis de nvmeris integris conferre, qvae in partibvs prioribvs tractata svnt.
Effectvs in parte demonstratos ab iis distingvere qvi admissi svnt et in aliis cvrsibvs demonstrabvntvr.
Problemata solvere qvae plvres lectiones partis conivngvnt et ideam non mechanicam postvlant, inter qvae est formatio rei concretae per polynomia.

Qvae vnitas consecvta est

Fvnctiones et earvm operationes

Pars a notione fvnctionis incepit: regvla qvae cviqve elemento dominii vnam omnino imaginem assignat. In lectione de fvnctionibvs algebraicis statvtae svnt regvlae qvae dominivm formvlae determinant, demonstratvm est lemma de signo prodvcti, fvndamentvm totivs posterioris tractationis signorvm, et fvnctiones in polynomiales, rationales, irrationales et transcendentes divisae svnt; divisionem, qvae ad fvnctiones, non ad earvm formvlas, pertinet, vacvam non esse ostendit demonstratio qva constat $\frac{1}{x}$ cvm nvllo polynomio congrvere, et lectio de algebra polynomiorvm postea demonstravit $\sqrt{x}$ fvnctionem rationalem non esse. Lectio de operationibvs fvnctionvm fvnctiones reales vt obiecta tractavit. Ex lectione de copiis, relationibvs et fvnctionibvs partis primae hereditate accepit associativitatem compositionis et theorema qvo fvnctio inversam habet si et solvm si biiectiva est; addidit qvae fvnctionvm realivm propria svnt: dominivm compositionis, qvo compositio associativa manet sed non commvtativa; fvnctionem stricte monotonam in copiam imaginvm svam biiectivam esse et inversam monotonam habere, qvod sinit fvnctiones non iniectivas per restrictionem invertere; delineationem inversae esse reflexionem ad rectam $y = x$ ; et omnem fvnctionem dominii symmetrici vno tantvm modo svmmam esse fvnctionis paris et fvnctionis imparis.

Polynomia et eorvm factorizatio

Centrvm partis est theoria polynomiorvm, qvae vt catena theorematvm evolvitvr, qvorvm qvodqve priori nititvr.

Algorithmvs divisionis, in lectione de algebra polynomiorvm demonstratvs, qvotientem et residvvm vnica praebet. Divisore $x - r$ , residvvm est $P(r)$ (theorema residvi), et inde theorema factoris: $r$ radix est si et solvm si $x - r$ dividit $P$ .
Ad plvres radices adhibitvm, theorema factoris praebet lemma de radicibvs diversis: polynomivm a nihilo differens gradvs $n$ svmmvm $n$ radices habet. Ex eo dedvcitvr principivm identitatis: dvo polynomia eandem fvnctionem definivnt si et solvm si eosdem coefficientes habent. Cvm vsqve ad id tempvs polynomia per coefficientes tractata essent, ordo non est circvlaris; et ex eo tempore qvodqve polynomivm cvm sva fvnctione identificatvr, coefficientes comparari possvnt, dvo polynomia qvae in infinitis pvnctis congrvvnt aeqvalia svnt, et ostenditvr radicem qvadratam fvnctionem rationalem non esse.
Ad radices inveniendas, lectio de factorizatione polynomii qvadratici qvadratvm complevit et formvlam qvadraticam, discriminantem et relationes Vietae obtinvit; lectio de radicibvs polynomiorvm gradvs svperioris divisionem syntheticam, theorema radicvm rationalivm, mvltiplicitatem et margines radicvm addidit.
Admisso theoremate fvndamentali algebrae, et cvm theoremate radicvm conivgatarvm, demonstrata est factorizatio in realibvs: omne polynomivm reale est coefficiens principalis eivs mvltiplicatvs per factores $x - r$ et per factores qvadraticos irredvcibiles.

P(x) = a_n (x - r_1) \cdots (x - r_k) \left( x^2 + p_1 x + s_1 \right) \cdots \left( x^2 + p_m x + s_m \right), \qquad k + 2m = n

A qvotiente ad delineationem

Factorizatione in manibvs, altervm dimidivm partis qvotientes consideravit. Lectio de fractionibvs algebraicis ostendit simplificationem factoris commvnis aeqvalitatem esse solvm in dominio originali, aeqvalitatem dvarvm fractionvm characterizavit et exsistentiam et vnicitatem decompositionis in fractiones partiales demonstravit cvm denominator solos factores lineares habet: factoribvs diversis, coefficientes obtinentvr aestimando in radicibvs identitatem qvae, per principivm identitatis, valet etiam vbi fractiones definitae non svnt; factoribvs iteratis, vnicitas postvlat vt in qvaqve radice terminvs potentiae maximae segregetvr. Lectio de tabvlis signorvm et inaeqvationibvs regvlam signorvm ad $n$ factores extendit et in factorizatione signvm polynomii vel qvotientis legit: signvm mvtatvr transevndo radicem mvltiplicitatis imparis, servatvr in radice mvltiplicitatis paris, et factores irredvcibiles non nvmerantvr. Inde secvta svnt solvtio inaeqvationvm et dominiorvm et distinctio inter asymptotos verticales et pvncta evitabilia.

Dvae vltimae lectiones omnia instrvmenta conivnxervnt. In lectione de fvnctionibvs rationalibvs, divisio $\frac{P}{Q}$ scribit vt polynomivm $C$ additvm fractioni propriae qvae ad nihilvm tendit, et $C$ est asymptotvs horizontalis, obliqva vel cvrvilinea; tabvla signorvm decernit qvo rami effvgiant, et discriminans aeqvationis $y \cdot Q(x) = P(x)$ copiam imaginvm praebet cvm ea aeqvatio gradvs $2$ est. In lectione de fvnctionibvs cvm radicalibvs, aeqvalitas $\sqrt{x^2} = \abs{x}$ cogit vt valores magni positivi et negativi variabilis seorsvm tractentvr, et vna eademqve fvnctio dvas asymptotos diversas habere potest.

Qvae admissa svnt

Pars aperte dixit qvae non demonstravit. Admissvm est theorema fvndamentale algebrae, cvivs demonstrationes instrvmenta analyseos postvlant. Sagittae limitvm, qvibvs asymptoti et habitvs fvnctionis prope pvnctvm vel in infinito describvntvr, sensv intvitivo adhibitae svnt: demonstratvm est signvm fvnctionis in vtroqve latere et decompositio qvae asymptotvm separat, descriptvs avtem ipse accessvs. Admissvm est fvnctiones rationales et fvnctiones cvm radicalibvs in qvoqve intervallo dominii svi continvas esse, et, forma generali, theorema valoris intermedii: fvnctio continva in intervallo $\icc{a}{b}$ omnem valorem inter $f(a)$ et $f(b)$ comprehensvm accipit, et praesertim evanescit si $f(a)$ et $f(b)$ signa contraria habent, qvod est theorema Bolzani; eo localizata svnt nihila qvae algebra non attingit et copiae imaginvm compvtatae svnt. Admissae svnt exsistentia et vnicitas decompositionis in fractiones partiales cvm factores qvadratici irredvcibiles adsvnt, qvae identitatem Bezovtianam inter polynomia postvlant, et vnicitas factorizationis in realibvs, ex qva pendet minimvm commvne mvltiplvm plvrivm polynomiorvm determinatvm esse praeter factorem constantem. Postremo admissvm est omnem fvnctionem expressione algebraica definitam aeqvationi polynomiali in $x$ et $y$ satisfacere, qvod factvm nvllvs alivs effectvs reqvirit. Et aperta manservnt maxima et minima qvae transformatione algebraica non revelantvr. Limites, continvitas, valor intermedivs, extrema et integratio fractionvm partialivm in cvrsibvs Calcvlvs I et Calcvlvs II tractantvr; theorema fvndamentale algebrae contra in cvrsv analyseos complexae demonstratvr.

Pictvra qvae seqvitvr tria segmenta in vnam catenam colligit. Qvisqve nodvs colvmnae est definitio avt theorema, secvndvm ordinem logicvm nvmeratvs et arcv cvm anvlis prioribvs conivnctvs qvibvs eivs demonstratio vtitvr; circvm flvitant fvlcra, id est principia et instrvmenta ad qvae catena itervm atqve itervm recvrrit, qvodqve linea continva cvm anvlis qvi ei nitvntvr conivnctvm. Omnia admittvntvr avt ex partibvs prioribvs provenivnt, et ideo vt circvli vacvi pingvntvr: effectvs qvos pars demonstrat et itervm adhibet, vt lemma de signo prodvcti vel theorema factoris, vt anvli apparent, arcvbvs cvm iis conivncti qvi eis vtvntvr. Fvlcro pvlsato omnes anvli qvi eo vtvntvr illvstrantvr, et qvisqve anvlvs ad lectionem remittit in qva demonstratvr.

Fvlcra catenae

AIndvctio. Proprietas qvae in casv initiali valet et a qvoqve casv ad evm qvi proxime seqvitvr transit, vel ab omnibvs prioribvs ad evm qvi proxime seqvitvr, semper valet; in parte de nvmeris natvralibvs demonstratvm. In hac vnitate sine demonstratione admittitvr.
BLemma Evclidis. Si nvmervs primvs prodvctvm nvmerorvm integrorvm dividit, aliqvem factorem dividit; inde seqvitvr, si nvmervs integer prodvctvm dividit et ad vnvm factorvm primvs est, evm altervm dividere; in parte de nvmeris natvralibvs demonstratvm. In hac vnitate sine demonstratione admittitvr.
CCompositio et inversa. Compositio fvnctionvm associativa est et iniectivitatem servat, et fvnctio inversam habet si et solvm si biiectiva est, eaqve inversa vnica est; in lectione de copiis, relationibvs et fvnctionibvs demonstratvm. In hac vnitate sine demonstratione admittitvr.
DRadices conivgatae. Si nvmervs complexvs radix est polynomii coefficientivm realivm, etiam eivs conivgatvs radix est; in parte de generibvs nvmerorvm demonstratvm. In hac vnitate sine demonstratione admittitvr.
ETheorema fvndamentale algebrae. Omne polynomivm gradvs $n \geq 1$ cvm coefficientibvs complexis saltem vnam radicem complexam habet; sine demonstratione admittitvr. In hac vnitate sine demonstratione admittitvr.
FBezovtivs et factorizatio vnica. Si dvo polynomia factores commvnes gradvs positivi non habent, exstant polynomia $U$ et $V$ cvm $U A + V B = 1$ ; et factorizatio polynomii realis in factores lineares et qvadraticos irredvcibiles vnica est praeter ordinem; admittvntvr, qvia cvrsvs maximvm commvnem divisorem polynomiorvm non evolvit. In hac vnitate sine demonstratione admittitvr.
GLimites et continvitas. Sagittae $x \to r$ et $f(x) \to \infty$ sensvm intvitivvm habent, et fvnctiones rationales et fvnctiones cvm radicalibvs in qvoqve intervallo dominii svi continvae svnt; admittitvr vsqve ad cvrsvm Calcvli I. In hac vnitate sine demonstratione admittitvr.
HValor intermedivs. Si fvnctio in intervallo $\icc{a}{b}$ continva est, in eo omnem valorem inter $f(a)$ et $f(b)$ comprehensvm accipit; praesertim, si $f(a)$ et $f(b)$ signa contraria habent, in aliqvo pvncto intervalli $\ioo{a}{b}$ evanescit (theorema Bolzani); admittitvr vsqve ad cvrsvm Calcvli I. In hac vnitate sine demonstratione admittitvr.

Segmentvm 1 Fvnctiones et earvm operationes
- Fvnctionem definire et regvlas statvere qvae dominivm formvlae determinant.
- Lemma de signo prodvcti demonstrare, fvndamentvm posterioris tractationis signorvm.
- Fvnctiones algebraicas dividere et ostendere divisionem vacvam non esse.
- Fvnctiones reales vt obiecta tractare: dominivm compositionis, inversa per restrictionem ad copiam in qva fvnctio monotona est, delineatio inversae et decompositio in partes parem et imparem.
1. Fvnctio et dominivm. Fvnctio cviqve elemento dominii svi vnam omnino imaginem assignat; dominivm formvlae exclvdit nihila denominatorvm et radicanda negativa radicvm indicis paris.
  Demonstratvr in: Fvnctiones algebraicae variabilis realis.
2. Signvm prodvcti. Pro nvmeris realibvs $a$ et $b$ , $ab = 0$ si et solvm si $a = 0$ vel $b = 0$ , et $ab > 0$ si et solvm si ambo a nihilo differvnt et eivsdem signi svnt; si $b \neq 0$ , qvotiens $\frac{a}{b}$ signvm habet ipsivs $ab$ .
  Demonstratvr in: Fvnctiones algebraicae variabilis realis.
3. Qvotiens qvi polynomivm non est. Nvlla fvnctio polynomialis cvm $\frac{1}{x}$ in omnibvs nvmeris realibvs a nihilo differentibvs congrvit; vnde exstant fvnctiones rationales qvae polynomiales non svnt.
  Vtitvr: 1. Fvnctio et dominivm. Demonstratvr in: Fvnctiones algebraicae variabilis realis.
4. Delineationes potentiarvm et radicvm. Pro $n \geq 2$ , fvnctiones $x^n$ et $\sqrt[n]{x}$ ordinem nvmerorvm realivm non negativorvm servant, $x^n < x < \sqrt[n]{x}$ si $0 < x < 1$ , et potentia par vel impar est provt $n$ par vel impar est.
  Vtitvr: 1. Fvnctio et dominivm. Demonstratvr in: Fvnctiones algebraicae variabilis realis.
5. Dominivm compositionis. Pro fvnctionibvs realibvs, dominivm fvnctionis $f \circ g$ constat ex iis $x$ dominii fvnctionis $g$ qvorvm imago in dominio fvnctionis $f$ est; his dominiis compositio associativa est, nec commvtativa.
  Vtitvr: 1. Fvnctio et dominivm. Nititvr: C (Compositio et inversa). Demonstratvr in: Operationes fvnctionvm, inversa et transformatio delineationvm.
6. Fvnctiones monotonae. Omnis fvnctio stricte crescens vel stricte decrescens iniectiva est, et svmma dvarvm fvnctionvm crescentivm crescens est.
  Vtitvr: 1. Fvnctio et dominivm. Demonstratvr in: Operationes fvnctionvm, inversa et transformatio delineationvm.
7. Inversa fvnctionis monotonae. Fvnctio stricte crescens in copia $A$ nvmerorvm realivm biiectiva est ex $A$ in copiam imaginvm svam, et eivs inversa stricte crescens est; idem valet de stricte decrescentibvs. Sic fvnctiones non iniectivae, dominio restricto, invertvntvr.
  Vtitvr: 6. Fvnctiones monotonae. Nititvr: C (Compositio et inversa). Demonstratvr in: Operationes fvnctionvm, inversa et transformatio delineationvm.
8. Delineatio inversae. Pvnctvm $(a, b)$ in delineatione fvnctionis $f$ est si et solvm si $(b, a)$ in ea fvnctionis $f^{-1}$ est; vnde delineatio inversae est reflexio delineationis fvnctionis $f$ ad rectam $y = x$ .
  Vtitvr: 7. Inversa fvnctionis monotonae. Nititvr: C (Compositio et inversa). Demonstratvr in: Operationes fvnctionvm, inversa et transformatio delineationvm.
9. Partes par et impar. Prodvctvm dvarvm fvnctionvm parivm vel dvarvm imparivm par est, prodvctvm avtem paris et imparis impar est; omnis fvnctio dominii symmetrici vno tantvm modo svmma est fvnctionis paris et fvnctionis imparis.
  Vtitvr: 1. Fvnctio et dominivm. Demonstratvr in: Operationes fvnctionvm, inversa et transformatio delineationvm.
Segmentvm 2 Polynomia et eorvm factorizatio
- Algorithmvm divisionis demonstrare et ex eo theoremata residvi et factoris dedvcere.
- Nvmervm radicvm polynomii circvmscribere et principivm identitatis obtinere.
- Radices invenire: formvla qvadratica, discriminans, relationes Vietae, divisio synthetica, radices rationales, mvltiplicitas et margines.
- Demonstrare, admisso theoremate fvndamentali algebrae, factorizationem omnis polynomii realis in factores lineares et qvadraticos irredvcibiles.
1. Algorithmvs divisionis. Datis dvobvs polynomiis $P$ et $Q$ , cvm $Q$ a nihilo differente, exstant qvotiens $C$ et residvvm $R$ vnica talia vt $P = Q \cdot C + R$ , cvm $R$ nvllo vel gradvs minoris qvam gradvs polynomii $Q$ .
  Nititvr: A (Indvctio). Demonstratvr in: Algebra polynomiorvm realivm.
2. Theoremata residvi et factoris. Residvvm divisionis polynomii $P$ per $x - r$ est $P(r)$ ; itaqve $x - r$ dividit $P$ si et solvm si $P(r) = 0$ .
  Vtitvr: 10. Algorithmvs divisionis. Demonstratvr in: Algebra polynomiorvm realivm.
3. Lemma de radicibvs diversis. Si $r_1, \ldots, r_k$ radices diversae polynomii $P$ svnt, tvm $P = (x - r_1) \cdots (x - r_k) \, C$ ; qvare polynomivm a nihilo differens gradvs $n$ svmmvm $n$ radices reales habet.
  Vtitvr: 2. Signvm prodvcti, 11. Theoremata residvi et factoris. Nititvr: A (Indvctio). Demonstratvr in: Algebra polynomiorvm realivm.
4. Principivm identitatis. Dvo polynomia in omni nvmero reali evndem valorem accipivnt si et solvm si eosdem coefficientes habent, potentiam pro potentia; vnde qvodqve polynomivm cvm sva fvnctione identificatvr, et radix qvadrata fvnctio rationalis non est.
  Vtitvr: 12. Lemma de radicibvs diversis. Demonstratvr in: Algebra polynomiorvm realivm.
5. Formvla qvadratica et discriminans. Qvadrato completo, radices polynomii $ax^2 + bx + c$ svnt $\frac{-b \pm \sqrt{\Delta}}{2a}$ , cvm $\Delta = b^2 - 4ac$ : dvae svnt si $\Delta > 0$ , vna si $\Delta = 0$ , nvlla si $\Delta < 0$ , qvo casv polynomivm in realibvs irredvcibile est.
  Vtitvr: 2. Signvm prodvcti, 11. Theoremata residvi et factoris. Demonstratvr in: Factorizatio polynomii qvadratici et trinomii gradvs 2n.
6. Factorizatio et relationes Vietae. Si $\Delta \geq 0$ , tvm $ax^2 + bx + c = a(x - x_1)(x - x_2)$ , cvm $x_1 + x_2 = -\frac{b}{a}$ et $x_1 x_2 = \frac{c}{a}$ .
  Vtitvr: 14. Formvla qvadratica et discriminans, 13. Principivm identitatis. Demonstratvr in: Factorizatio polynomii qvadratici et trinomii gradvs 2n.
7. Biqvadraticvm discriminantis negativi. Si $b^2 - 4ac < 0$ , trinomivm $ax^4 + bx^2 + c$ est $a$ mvltiplicatvm per prodvctvm dvorvm polynomiorvm qvadraticorvm realivm irredvcibilivm, $x^2 - kx + m$ et $x^2 + kx + m$ .
  Vtitvr: 14. Formvla qvadratica et discriminans. Demonstratvr in: Factorizatio polynomii qvadratici et trinomii gradvs 2n.
8. Divisio synthetica. Coefficientes qvotientis polynomii $P$ per $x - r$ obtinentvr coefficientem principalem demittendo et cviqve coefficienti addendo $r$ vicibvs nvmervm priorem; vltimvs nvmervs est residvvm, $P(r)$ .
  Vtitvr: 10. Algorithmvs divisionis, 13. Principivm identitatis. Demonstratvr in: Radices polynomiorvm gradvs svperioris et divisio synthetica.
9. Radices rationales. Si fractio irredvcibilis $\frac{p}{q}$ radix est polynomii coefficientivm integrorvm cvm $a_0 \neq 0$ , tvm $p$ dividit $a_0$ et $q$ dividit $a_n$ ; demonstratio lemmate Evclidis partis primae vtitvr.
  Nititvr: B (Lemma Evclidis). Demonstratvr in: Radices polynomiorvm gradvs svperioris et divisio synthetica.
10. Divisio synthetica iterata. Radix $r$ mvltiplicitatem $k$ habet, id est $P = (x - r)^k C$ cvm $C(r) \neq 0$ , si et solvm si, cvm deinceps per $x - r$ dividitvr, priora $k$ residva nvlla svnt et qvod proxime seqvitvr non est.
  Vtitvr: 17. Divisio synthetica, 10. Algorithmvs divisionis, 11. Theoremata residvi et factoris. Demonstratvr in: Radices polynomiorvm gradvs svperioris et divisio synthetica.
11. Margines radicvm. Omnis radix realis satisfacit $\abs{r} < 1 + \frac{M}{\abs{a_n}}$ (margo Cavchyanvs), vbi $M$ est maximvs valorvm absolvtorvm ceterorvm coefficientivm, et si divisio synthetica per $x - c$ , cvm $c > 0$ et $a_n > 0$ , versvm sine nvmeris negativis praebet, nvlla radix realis svperat $c$ .
  Vtitvr: 17. Divisio synthetica. Demonstratvr in: Radices polynomiorvm gradvs svperioris et divisio synthetica.
12. Factorizatio in realibvs. Omne polynomivm reale gradvs $n \geq 1$ est coefficiens principalis eivs mvltiplicatvs per factores $x - r$ , cvm $r$ reali, et per factores qvadraticos monicos in realibvs irredvcibiles.
  Vtitvr: 14. Formvla qvadratica et discriminans, 10. Algorithmvs divisionis, 17. Divisio synthetica. Nititvr: A (Indvctio), E (Theorema fvndamentale algebrae), D (Radices conivgatae). Demonstratvr in: Radices polynomiorvm gradvs svperioris et divisio synthetica.
Segmentvm 3 A qvotiente ad delineationem
- Accvrate definire qvando simplificatio aeqvalitas sit et qvando dvae fractiones algebraicae aeqvales sint.
- Fractionem propriam in fractiones partiales decomponere.
- In factorizatione signvm polynomii vel qvotientis legere et eo inaeqvationes et dominia solvere.
- Fvnctiones rationales et fvnctiones cvm radicalibvs examinare: asymptoti verticales, horizontales, obliqvae et cvrvilineae, fines dominii et copia imaginvm.
1. Lemma simplificationis. In omni nvmero reali qvi $QC$ non annvllat, fractio $\frac{PC}{QC}$ idem valet ac $\frac{P}{Q}$ ; ambae eadem fvnctio svnt solvm si altera ad dominivm prioris restringitvr.
  Vtitvr: 2. Signvm prodvcti. Demonstratvr in: Fractiones algebraicae et fractiones partiales.
2. Aeqvalitas fractionvm. Dvo polynomia qvae in omnibvs nvmeris realibvs praeter nvmervm finitvm congrvvnt aeqvalia svnt; inde seqvitvr $\frac{P}{Q}$ et $\frac{R}{S}$ vbi $QS$ non evanescit congrvere si et solvm si $PS = QR$ .
  Vtitvr: 12. Lemma de radicibvs diversis, 13. Principivm identitatis. Demonstratvr in: Fractiones algebraicae et fractiones partiales.
3. Fractiones partiales. Omnis fractio propria vnico modo svmma est terminorvm $\frac{A}{(x - r)^j}$ et $\frac{Bx + C}{(x^2 + px + q)^j}$ , vnivs pro qvaqve potentia cvivsqve factoris denominatoris svi; factoribvs linearibvs, exsistentia obtinetvr singvlis terminis separandis et vnicitas in qvaqve radice maxima potentia segreganda; factoribvs qvadraticis irredvcibilibvs, ambae admittvntvr.
  Vtitvr: 2. Signvm prodvcti, 11. Theoremata residvi et factoris, 10. Algorithmvs divisionis, 23. Aeqvalitas fractionvm, 21. Factorizatio in realibvs. Nititvr: F (Bezovtivs et factorizatio vnica). Demonstratvr in: Fractiones algebraicae et fractiones partiales.
4. Signvm polynomii. Prodvctvm $n$ factorvm a nihilo differentivm negativvm est si nvmervm imparem factorvm negativorvm habet; itaqve signvm polynomii in factores resolvti mvtatvr transevndo radicem si et solvm si eivs mvltiplicitas impar est, et factores irredvcibiles signvm coefficientis principalis svi servant.
  Vtitvr: 2. Signvm prodvcti, 14. Formvla qvadratica et discriminans, 19. Divisio synthetica iterata. Nititvr: A (Indvctio). Demonstratvr in: Tabvlae signorvm et inaeqvationes.
5. Asymptoti verticales et pvncta evitabilia. Si $r$ denominatorem fractionis $\frac{A}{B}$ annvllat et $A(r) \neq 0$ , recta $x = r$ est asymptotvs verticalis; si $A(r) = 0$ , factor commvnis tollitvr et, si fractio simplificata in $r$ definita est, hoc est pvnctvm evitabile.
  Vtitvr: 11. Theoremata residvi et factoris, 22. Lemma simplificationis, 25. Signvm polynomii. Nititvr: G (Limites et continvitas). Demonstratvr in: Tabvlae signorvm et inaeqvationes.
6. Asymptoti fvnctionis rationalis. Si $P = Q \cdot C + R$ , differentia inter $\frac{P}{Q}$ et $C$ est $\frac{R}{Q}$ , qvae in vtroqve termino ad $0$ tendit: qvotiens $C$ est asymptotvs horizontalis, obliqva vel cvrvilinea, et tabvla signorvm decernit qvo rami prope verticales effvgiant.
  Vtitvr: 10. Algorithmvs divisionis, 26. Asymptoti verticales et pvncta evitabilia. Nititvr: G (Limites et continvitas). Demonstratvr in: Fvnctiones rationales: asymptoti et delineatio.
7. Nihila et copia imaginvm fvnctionis rationalis. Per valorem intermedivm, fvnctio rationalis qvae in intervallo dominii svi signvm mvtat in eo evanescit, et fvnctio qvae in intervallo dominii svi valores qvovis nvmero maiores et minores accipit ibi omnes valores reales accipit; si aeqvatio $y \cdot Q(x) = P(x)$ gradvs $2$ est, eivs discriminans copiam imaginvm praebet.
  Vtitvr: 14. Formvla qvadratica et discriminans, 27. Asymptoti fvnctionis rationalis. Nititvr: G (Limites et continvitas), H (Valor intermedivs). Demonstratvr in: Fvnctiones rationales: asymptoti et delineatio.
8. Dominivm et fines cvm radicalibvs. Dominivm fvnctionis cvm radicalibvs est intersectio copiarvm qvas tabvlae signorvm condicionvm eivs praebent; si radix mvltiplicitatis imparis radicandvm nvmeratoris annvllat, delineatio ibi in axe $x$ finitvr; si denominator evanescit nvmeratore a nihilo differente, adest asymptotvs verticalis; et, per valorem intermedivm, qvisqve ramvs omnes valores accipit inter eos comprehensos qvos in finibvs svis attingit vel ad qvos accedit.
  Vtitvr: 25. Signvm polynomii, 26. Asymptoti verticales et pvncta evitabilia. Nititvr: G (Limites et continvitas), H (Valor intermedivs). Demonstratvr in: Fvnctiones cvm radicalibvs: dominivm, signa et delineatio.
9. Asymptoti cvm radicalibvs. Cvm $\sqrt{x^2} = \abs{x}$ , asymptotvs fvnctionis cvm radicalibvs seorsvm invenitvr cvm $x \to \infty$ et cvm $x \to -\infty$ , et in vtroqve termino diversa esse potest; si $\frac{f(x)}{x}$ ad nvmervm non tendit, nvlla est asymptotvs recta.
  Vtitvr: 27. Asymptoti fvnctionis rationalis. Nititvr: G (Limites et continvitas). Demonstratvr in: Fvnctiones cvm radicalibvs: dominivm, signa et delineatio.

Ad geometriam analyticam. Pars seqvens cvrvas plani per earvm aeqvationes considerat: rectas, parabolam, qvae qvadrato complendo characterizatvr, circvmferentiam vt vnionem delineationvm fvnctionvm $\pm \sqrt{r^2 - x^2}$ et hyperbolam $y = \frac{1}{x}$ cvm dvabvs eivs asymptotis.