❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❚❡♦r❡♠❛ ❞❡ ▼❛r❞❡♥
▼ár✐♦ ❏♦♥❛s ❞❛ ❙✐❧✈❛ ❙❛♥t♦s
●♦✐â♥✐❛
▼ár✐♦ ❏♦♥❛s ❞❛ ❙✐❧✈❛ ❙❛♥t♦s
❚❡♦r❡♠❛ ❞❡ ▼❛r❞❡♥
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❍❡♥r✐q✉❡
●♦✐â♥✐❛
Ficha catalográfica elaborada
Automaticamente com os dados fornecidos pelo(a) autor(a).
Santos, Mário Jonas da Silva.
Teorema de Marden [manuscrito] / Mário Jonas da Silva Santos. - 2014.
lxi, 61 f.
Orientador: Prof. Dr. Paulo Henrique de Azevedo Rodrigues.
Dissertação (Mestrado) – Universidade Federal de Goiás, Instituto de Matemática e Estatística, Programa de Pós-Graduação em Matemática, Goiânia, 2014.
Bibliografia.
Inclui siglas, tabelas, inclui lista de figuras.
1.Teorema de Marden 2.Polinômio com coeficientes complexos. I. Rodrigues, Dr. Paulo Henrique de Azevedo, Orient. II. Titulo
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❞♦ ❛✉t♦r ▼ár✐♦ ❏♦♥❛s ❞❛ ❙✐❧✈❛ ❙❛♥t♦s ❡ ❞♦ ♦r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉r✳ P❛✉❧♦ ❍❡♥r✐q✉❡✳
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s
q✉❡ s❡♠♣r❡ ❡st✐✈❡r❛♠ ❞♦ ♠❡✉ ❧❛❞♦ ♠❡ ❞❛❞♦ ♦ s✉♣♦rt❡
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ♣❡❧❛ ♠✐♥❤❛ ✈✐❞❛✱ s❛ú❞❡ ❡ t❡r ♠❡ ❛❜❡♥ç♦❛❞♦ ♣❛r❛ ❝❤❡❣❛r ♥❡ss❡ ♠♦♠❡♥t♦✳ ▼❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉r✳ P❛✉❧♦ ❍❡♥r✐q✉❡ ♣❡❧❛ ❞❡❞✐❝❛çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ♦r✐❡♥t❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ Pr♦❢❡ss♦r✦✦ ❚❛♠❜é♠ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ❝♦♥tr✐❜✉✐r❛♠ ❝♦♠ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♣r♦✜ss✐♦♥❛❧✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦ ♣r♦❢❡ss♦r ❱❛❧❞✐✈✐♥♦ q✉❡ ♠✐♥✐str♦✉ ❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❊s✲ t❛tíst✐❝❛ ❝♦♠ ♠✉✐t❛ ❞❡❞✐❝❛çã♦✱ ❡ ✐st♦ ♠❡ ♣r♦♣♦r❝✐♦♥♦✉ ✉♠ ❛♣r✐♠♦r❛♠❡♥t♦ ❡♠ ♠❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ♦ r❡s✉❧t❛❞♦ ❞✐ss♦ ❢♦✐ ❛ ♠✐♥❤❛ ❛♣r♦✈❛çã♦ ❡♠ ✉♠ ❝♦♥❝✉rs♦ ♣✉❜❧✐❝♦ ♥❛ ❛r❡❛ ❞❡ ❊st❛tíst✐❝❛ ❡ Pr♦❜❛❜✐❧✐❞❛❞❡✳
◗✉❡r♦ ❛❣r❛❞❡ç❡r ❛ ♠✐♥❤❛ ❡s♣♦s❛✱ ❊st❡r✱ ♣❡❧❛ ❛❥✉❞❛✱ ❝♦♠♣r❡❡♥sã♦✱ ❝❛r✐♥❤♦ ❡ ♣❛❝✐✲ ê♥❝✐❛ ❞❡♠♦♥str❛❞❛ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❛♥♦s✳ ❚❛♠❜é♠ ❛ ♠❡✉s ♣❛✐s ❈❧❡✐❞❡♠❛r ❡ ❆♣❛r❡❝✐❞❛✳ ▼❡ ❞❡s❝✉❧♣❡ ♣❡❧♦s ♠♦♠❡♥t♦s q✉❡ t✐✈❡ q✉❡ ♠❡ ❛✉s❡♥t❛r ♣❛r❛ ❡st✉❞❛r✳ ❱♦❝ês sã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ♥❛ ♠✐♥❤❛ ✈✐❞❛✳
✾
❘❡s✉♠♦
❱❛♠♦s ❝♦♠❡ç❛r ❢❛③❡♥❞♦ ✉♠❛ ❡①♣❧❛♥❛çã♦ ❞❡ ❛❧❣✉♥s ❝♦♥t❡ú❞♦s ✐♠♣♦rt❛♥t❡s✳ ❈♦✲ ♠❡ç❛♥❞♦ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ♣♦❧✐♥ô♠✐♦s✱ ❡❧✐♣s❡✱ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♥❛ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛ ❡ ❝♦♥❣✉ê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s ❡♠ s❡❣✉✐❞❛ ✈❛♠♦s ❡♥✉❝✐❛r três ❧❡♠❛s ❡ ❞❡♠♦♥strá✕❧♦s ♣❛r❛ ❡♥tã♦ ❡♥✉♥❝✐❛r ❡ ❞❡♠♦♥str❛r ♦ ❚❡♦r❡♠❛ ❞❡ ▼❛r❞❡♥✳ ❆♦ ✜♥❛❧ t❡r❡♠♦s ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❛✉❧❛ ❡♠ ❢♦r♠❛ ❞❡ ♦✜❝✐♥❛ ♠❛t❡♠át✐❝❛✱ ❛♣❧✐❝❛❞❛ ♣❛r❛ ❛❧✉♥♦s ❞❛ ✸❛ ❙ér✐❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡
✶✵
❆❜str❛❝t
▲❡t✬s st❛rt ♠❛❦✐♥❣ ❛♥ ❡①♣❧❛♥❛t✐♦♥ ♦❢ s♦♠❡ ✐♠♣♦rt❛♥t ❝♦♥t❡♥t✳ ❙t❛rt✐♥❣ ✇✐t❤ t❤❡ s❡t ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ♣♦❧②♥♦♠✐❛❧s✱ ❡❧❧✐♣s❡✱ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡ ❛♥❞ ❝♦♥❣r✉❡♥❝❡ tr✐❛♥❣❧❡s t❤❡♥ ✇❡ ❡♥✉♥❝✐❛t❡ t❤r❡❡ ❧❡♠♠❛ ❛♥❞ ❞❡♠♦♥str❛t❡s ❢♦r t❤❡♠ t❤❡♥ ❡♥✉♥❝✐❛t❡ ❛♥❞ ♣r♦✈❡ ❚❤❡♦r❡♠ ▼❛r❞❡♥✳ ❆t t❤❡ ❡♥❞ ✇❡ ✇✐❧❧ ❤❛✈❡ ❛ ♣r♦♣♦s❛❧ t♦ ❝❧❛ss ✐♥ t❤❡ ❢♦r♠ ♦❢ ♠❛t❤ ✇♦r❦s❤♦♣✱ st✉❞❡♥ts ❛♣♣❧✐❡❞ ❢♦r t❤❡ ✸r❞ ❙❡r✐❡s ♦❢ ❤✐❣❤ s❝❤♦♦❧✳
❑❡②✇♦r❞s
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ P❧❛♥♦ ❈❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❉✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸ P❧❛♥♦ ❈❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✹ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✺ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✻ ❋♦r♠❛ ❚r✐❣♦♥♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✼ ❊①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✽ ■♥t❡r♣r❡t❛çã♦ ●❡♦♠étr✐❝❛ ❞❛ ❉❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✾ ❚r✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✶✵ ❊①❡♠♣❧♦ ❞❡ tr✐â♥❣✉❧♦s ❝♦♥❣r✉ê♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✶✶ ❈❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✶✷ ▼❡❞✐❛♥❛✱ ❜✐ss❡tr✐③ ❡ ❛❧t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✶✸ ❈❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ▲▲▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸✳✶ ❈♦♥✜❣✉r❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼❛r❞❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷ ❊❧✐♣s❡✱ r❡t❛s t❛♥❣❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✸ ❚r✐â♥❣✉❧♦s ✐sós❝❡❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✹ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✺ ❚r✐â♥❣✉❧♦ ❝♦♠ ✈ért✐❝❡ ♥❛ ♦r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✻ ❊❧✐♣s❡ t❛♥❣❡♥t❡ ♥♦s ♣♦♥t♦s ♠é❞✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✹✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
❙✉♠ár✐♦
✶ ■◆❚❘❖❉❯➬➹❖ ✶✹
✷ P❘❊▲■▼■◆❆❘❊❙ ✶✻
✷✳✶ ❈❖❖❘❉❊◆❆❉❆❙ ◆❖ P▲❆◆❖ ❈❆❘❚❊❙■❆◆❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✳✶ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❊▲■P❙❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ ◆Ú▼❊❘❖❙ ❈❖▼P▲❊❳❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✷ ❆ ❋♦r♠❛ ❆❧❣é❜r✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸✳✸ ❆ ❋♦r♠❛ ❚r✐❣♦♥♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ❊◗❯❆➬Õ❊❙ P❖▲■◆❖▼■❆■❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✹✳✶ P♦❧✐♥ô♠✐♦s ❈♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✹✳✷ ❉✐✈✐sã♦ ❞❡ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✹✳✸ ❉✐✈✐sã♦ ❞❡ ✉♠ P♦❧✐♥ô♠✐♦ ♣♦r (x−α) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹✳✹ ❖ ❚❡♦r❡♠❛ ❋✉♠❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✺ ❉❊❘■❱❆❉❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺✳✶ ❆ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❋✉♥çã♦ ❞❡ ❱❛r✐á✈❡❧ ❘❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺✳✷ ■♥t❡r♣r❡t❛çã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✻ ❈❖◆●❘❯✃◆❈■❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✻✳✶ ❈❛s♦s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸ ❖ ❚❊❖❘❊▼❆ ✹✹
✸✳✶ ❖ ❚❊❖❘❊▼❆ ❉❊ ▼❆❘❉❊◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹ ❖❋■❈■◆❆ ✺✶
✹✳✶ ❈✉rrí❝✉❧♦ ❞❡ ❘❡❢❡rê♥❝✐❛ ❞❛ ❘❡❞❡ ❊st❛❞✉❛❧ ❞❡ ❊❞✉❝❛çã♦ ❞❡ ●♦✐ás ✳ ✳ ✳ ✳ ✺✶
❙❯▼➪❘■❖ ✶✸
✹✳✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ❉❡s❛♣r❡♥❞✐③❛❣❡♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✶✳✷ ❆✉❧❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✶✳✸ ❆✉❧❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✶✳✹ ❆✉❧❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✶✳✺ ❆✉❧❛ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
❈❛♣ít✉❧♦ ✶
■◆❚❘❖❉❯➬➹❖
❖❜s❡r✈❛♥❞♦ q✉❡ ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ❛✐♥❞❛ ❢✉♥❝✐♦♥❛ ♥❛ ❢♦r♠❛ ❞♦ ✧❞❡❝♦r❡❜❛✧✱ ♦✉ s❡❥❛✱ ♦ ♣r♦❢❡ss♦r ❢❛③ ❡①❡♠♣❧♦s ❡ ♣r♦♣õ❡ ✉♠❛ sér✐❡ ❞❡ ❡①❡r❝í❝✐♦s ♣❛r❡❝✐❞♦s ♣❛r❛ q✉❡ ♦s ❛❧✉♥♦s ♠❡♠♦r✐③❡♠ ❛ ❢ór♠✉❧❛ ♦✉ ♦ ❝♦♥t❡ú❞♦ ❡♥s✐♥❛❞♦✳ P❡♥s❛♥❞♦ ♥❡st❡ ❢❛t♦✱ ❡ t❡♥t❛♥❞♦ ❡♥❝♦♥tr❛r ✉♠❛ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ❞❡ ❡♥s✐♥❛r ❛❧❣✉♥s ❝♦♥t❡ú❞♦s ❡ t♦r♥❛♥❞♦✲♦s ♠❛✐s s✐❣♥✐✜❝❛t✐✈♦s✱ r❡❛❧✐③❛♠♦s ❡st❡ tr❛❜❛❧❤♦✳ ◗✉❡ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❞✐s❝✉t✐r ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s r❛í③❡s ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ❞❡ ❣r❛✉ ✸ ❡ ❛s r❛í③❡s ❞❡ s✉❛ ❞❡r✐✈❛❞❛✳
❊♠ ♠❛t❡♠át✐❝❛✱ ❛s ✐♥❞❛❣❛çõ❡s ✐♥st✐❣❛♠ ♦s ❛❧✉♥♦s✱ ❡ t❛♠❜é♠ ♣r♦❢❡ss♦r❡s✱ ❛ ❡①✲ ♣❛♥❞✐r❡♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ❛ ❜✉s❝❛r❡♠ ❝♦♥❤❡❝✐♠❡♥t♦s ♥♦✈♦s ❛ r❡s♣❡✐t♦ ❞♦ t❡♠❛ ❡♥s✐♥❛❞♦✱ ❡ ❡ss❛ ❜✉s❝❛ t♦r♥❛ ♦ ♣r♦❝❡ss♦ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ♠✉✐t♦ ♠❛✐s ♣r❛③❡r♦s♦ ❡ ♣r♦✈❡✐t♦s♦✳ P❡♥s❛♥❞♦ ♥✐ss♦✱ ❡ss❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❛ s❡❣✉✐♥t❡ ✐♥❞❛✲ ❣❛çã♦✿ ❈♦♥s✐❞❡r❡ ✉♠ ♣♦❧✐♥ô♠✐♦q✱ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s✱ ❞❡ ✸♦ ❣r❛✉✱ ❝✉❥❛s r❛í③❡s
sã♦ ♥ã♦ ❝♦❧✐♥❡❛r❡s✱ ♦✉ s❡❥❛✱ ❛s r❛í③❡s sã♦ ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦T✳ ❖ q✉❡ ♣♦❞❡♠♦s
❞✐③❡r ❞❛s r❛í③❡s ❞❛ ❞❡r✐✈❛❞❛ ❞❡ss❡ ♣♦❧✐♥ô♠✐♦❄ ❚♦❞❛ ❛ ❛❜♦r❞❛❣❡♠ ❞♦ tr❛❜❛❧❤♦ é ❡♠ t♦r♥♦ ❞❡ss❛ t❡♠át✐❝❛✱ ❛❣r✉♣❛♥❞♦ t♦❞♦s ♦s ♣r❡ss✉♣♦st♦s t❡ór✐❝♦s ❡ ❡♣st❡♠♦❧ó❣✐❝♦s ♣❛r❛ q✉❡ ❡ss❛ q✉❡stã♦ s❡❥❛ r❡s♣♦♥❞✐❞❛✱ ❡ ♠❛✐s✱ ❛♣r❡s❡♥t❛♥❞♦ s✉❣❡stõ❡s ❞❡ ❛♣❧✐❝❛çõ❡s ❡ ❞❡ ❛✉❧❛s ♣❛r❛ q✉❡ ❡ss❡ ❛ss✉♥t♦ s❡❥❛ ❛❜♦r❞❛❞♦ ♥❛ ❡s❝♦❧❛ ❝♦♠ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳
❆ q✉❡stã♦ ❧❡✈❛♥t❛❞❛ ♥♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r é ♥♦ ♠í♥✐♠♦ ❝✉r✐♦s❛✱ ❡ ❞❡s♣❡rt❛ ❞❡ ✐♠❡❞✐❛t♦ ♦ ❞❡s❡❥♦ ❞❡ ✐♥✈❡st✐❣❛çã♦ s♦❜r❡ ♦ ❛ss✉♥t♦✳ ❖ q✉❡ s❡ ✈❡r✐✜❝❛ é q✉❡ ❛s r❛í③❡s ❞♦ ♣♦❧✐♥ô♠✐♦qsã♦ ♣♦♥t♦s ✐♥t❡r♥♦s ❛♦ tr✐â♥❣✉❧♦T✱ ❡ss❡s ♣♦♥t♦s sã♦ ♦s ❢♦❝♦s ❞❡ ✉♠❛ ❡❧✐♣s❡ E✐♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦T✱ ❡ ♠❛✐s✱ ❡❧❛ t❛♥❣ê♥❝✐❛ ♦s ❧❛❞♦s ❞❡T ♥♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣♦♥t♦s
❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✶✺
♠é❞✐♦s✳ ❊ss❛ ❡❧✐♣s❡ é ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❊❧✐♣s❡ ❞❡ ❙t❡✐♥❡r ❡ ❡st❡ r❡s✉❧t❛❞♦ é ❞❡♥♦♠✐♥❛❞♦ ❞❡ ❚❡♦r❡♠❛ ❞❡ ▼❛r❞❡♥✳
❚♦❞♦ ♦ tr❛❜❛❧❤♦ ❢♦✐ ❡str✉t✉r❛❞♦ ♣❛r❛ ❞❡♠♦♥str❛r ♦ r❡s✉❧t❛❞♦ ❛♣r❡s❡♥t❛❞♦ ❛❝✐♠❛ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ▼❛r❞❡♥ ❡ ♣r♦♣♦r s✉❥❡stõ❡s ❞❡ ❛♣❧✐❝❛çõ❡s ❞♦ ❡st✉❞♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ P❛r❛ ✐st♦ ❛♣r❡s❡♥t❛♠♦s ♥♦ ❝❛♣ít✉❧♦ ❞♦✐s ❡st✉❞♦s ♣r❡❧✐♠✐♥❛r❡s s♦❜r❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ ♣❧❛♥♦✱ ❡❧✐♣s❡✱ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❡q✉❛çõ❡s ♣♦❧✐♥ô♠✐❛✐s ❡ s♦❜r❡ ❞❡r✐✈❛❞❛s✱ ♣❛r❛ ❞❛r ♦ s✉♣♦rt❡ t❡ór✐❝♦ ♣❛r❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛✳ ◆♦ ❝❛♣ít✉❧♦ três ❢❛③❡♠♦s ❛ ❞❡♠♦♥s✲ tr❛çã♦ ❞♦ t❡♦r❡♠❛ ✉s❛♥❞♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ✈♦❧t❛❞❛ ♣❛r❛ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❊ ❢❡❝❤❛♠♦s ♦ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♥❞♦ ♥♦ ❝❛♣ít✉❧♦ q✉❛tr♦ ♦✜❝✐♥❛s ♣❡❞❛❣ó❣✐❝❛s ❡ s✉❣❡stõ❡s ❞❡ ❛✉❧❛s s♦❜r❡ ❚❡♦r❡♠❛ ❞❡ ▼❛r❞❡♥✳
❈❛♣ít✉❧♦ ✷
P❘❊▲■▼■◆❆❘❊❙
✷✳✶ ❈❖❖❘❉❊◆❆❉❆❙ ◆❖ P▲❆◆❖ ❈❆❘❚❊❙■❆◆❖
❯♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✭❝❛rt❡s✐❛♥❛s✮ ♥♦ ♣❧❛♥♦ Π ❝♦♥s✐st❡ ♥✉♠ ♣❛r ❞❡ ❡✐①♦s ♣❡r✲ ♣❡♥❞✐❝✉❧❛r❡s OX ❡ OY ❝♦♥t✐❞♦s ♥❡ss❡ ♣❧❛♥♦✱ ❝♦♠ ❛ ♠❡s♠❛ ♦r✐❣❡♠ O✳ OX ❝❤❛♠❛✲s❡
♦ ❡✐①♦ ❞❛s ❛❜❝✐ss❛s ❡ OY é ♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✳ ❖ s✐st❡♠❛ é ✐♥❞✐❝❛❞♦ ❝♦♠ ❛ ♥♦t❛çã♦ OXY✳
❆ ❡s❝♦❧❤❛ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ ♣❧❛♥♦ Π ♣❡r♠✐t❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ Π→R2 ♦♥❞❡R2 é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s ♦r❞❡♥❛❞♦s (x, y)❞❡
♥ú♠❡r♦s r❡❛✐s✳ ❆ ❝❛❞❛ ♣♦♥t♦p❞♦ ♣❧❛♥♦Π ❝♦rr❡s♣♦♥❞❡ ✉♠ ♣❛r ♦r❞❡♥❛❞♦(x, y) ∈ R2✳
❖s ♥ú♠❡r♦s x ❡ y sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ s✐st❡♠ OXY✿ x
é ❛ ❛❜❝✐ss❛ ❡ y é ❛ ♦r❞❡♥❛❞❛ ❞❡ P✳ ❆s ❝♦♦r❞❡♥❛❞❛s x, y ❞♦ ♣♦♥t♦ P sã♦ ❞❡✜♥✐❞❛s ❞♦
s❡❣✉✐♥t❡ ♠♦❞♦✿
❙❡P ❡st✐✈❡r s♦❜r❡ ♦ ❡✐①♦OX✱ ♦ ♣❛r ♦r❞❡♥❛❞♦ q✉❡ ❧❤❡ ❝♦rr❡s♣♦♥❞❡ é(x,0)✱ ♦♥❞❡ xé
❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ P ♥♦ ❡✐①♦OX✳ ❙❡ P ❡st✐✈❡r s♦❜r❡ ♦ ❡✐①♦OY✱ ❛ ❡❧❡ ❝♦rr❡s♣♦♥❞❡ ♦ ♣❛r
(0, y)✱ ♦♥❞❡ y é ❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ P ♥❡ss❡ ❡✐①♦✳ ❙❡ P ♥ã♦ ❡stá ❡♠ q✉❛❧q✉❡r ❞♦s ❡✐①♦s✱
tr❛ç❛♠♦s ♣♦r P ✉♠❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ OY✱ ❛ q✉❛❧ ❝♦rt❛ OX ♥♦ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛ x ❡ ✉♠❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ OX✱ ❛ q✉❛❧ ❝♦rt❛ OY ♥♦ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛ y✳ ❊♥tã♦ x
s❡rá ❛ ❛❜❝✐ss❛ ❡ y ❛ ♦r❞❡♥❛❞❛ ❞♦ ♣♦♥t♦ P✳ ❖✉ s❡❥❛ (x, y) ∈ R2 é ♦ ♣❛r ♦r❞❡♥❛❞♦ ❞❡
♥ú♠❡r♦s r❡❛✐s q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♣♦♥t♦ P✳
❖ ♣♦♥t♦O✱ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ t❡♠ ❛❜❝✐ss❛ ❡ ♦r❞❡♥❛❞❛ ❛♠❜❛s ✐❣✉❛✐s
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✼
❋✐❣✉r❛ ✷✳✶✿ P❧❛♥♦ ❈❛rt❡s✐❛♥♦
❛ ③❡r♦✳ ❆ss✐♠✱ ❛ ❡❧❡ ❝♦rr❡s♣♦♥❞❡ ♦ ♣❛r (0,0) ∈ R2✳
❙❡ x é ❛ ❛❜❝✐ss❛ ❡ y é ❛ ♦r❞❡♥❛❞❛ ❞♦ ♣♦♥t♦ P✱ ♦ ♣♦♥t♦ P′ ❞❡ ❝♦♦r❞❡♥❛❞❛s
(x,0) ❝❤❛♠❛✲s❡ ❛ ♣r♦❥❡çã♦ ❞❡P s♦❜r❡ ♦ ❡✐①♦OX ❡♥q✉❛♥t♦ ♦ ♣♦♥t♦P′′✱ ❞❡ ❝♦♦r❞❡♥❛❞❛s
(0, y) é ❝❤❛♠❛❞♦ ❛ ♣r♦❥❡çã♦ ❞❡ P s♦❜r❡ ♦ ❡✐①♦ OY✳
❖ ❡♠♣r❡❣♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ ♣❧❛♥♦ s❡r✈❡ ❛ ❞♦✐s ♣r♦♣ós✐t♦s q✉❡ s❡ ❝♦♠♣❧❡♠❡♥t❛♠✳ ❖ ♣r✐♠❡✐r♦ é✱ ♦ ❞❡ ❛tr✐❜✉✐r ✉♠ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❛ ❢❛t♦s ❞❡ ♥❛t✉r❡③❛ ♥✉♠ér✐❝❛✱ ❝♦♠♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✱ q✉❡ ❣❛♥❤❛ ♠✉✐t♦ ❡♠ ❝❧❛r❡③❛ q✉❛♥❞♦ s❡ ♦❧❤❛ ♣❛r❛ s❡✉ ❣rá✜❝♦✳ ❖ s❡❣✉♥❞♦ ♣r♦♣ós✐t♦ ❞♦ ✉s♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ✈❛✐ ♥♦ s❡♥t✐❞♦ ♦♣♦st♦✿ r❡❝♦rr❡✲s❡ ❛ ❡❧❛s ❛ ✜♠ ❞❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❛ ●❡♦♠❡tr✐❛✳ ❊st❡ é ♦ ♦❜❥❡t✐✈♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ❛ ê♥❢❛s❡ r❡❝❛✐ s♦❜r❡ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ R2 →Π ❡ ♥♦ s❡❣✉♥❞♦ s♦❜r❡ s✉❛ ✐♥✈❡rs❛ Π → R2✳ ◆❛ ♣rát✐❝❛✱ ❡ss❡s
❞♦✐s ♣♦♥t♦s ❞❡ ✈✐st❛ s❡ ❡♥tr❡❧❛ç❛♠✿ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ♦s ❢❛t♦s ✐♥✐❝✐❛✐s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✉s❛♠✲s❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳
✷✳✶✳✶ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s
❙❡ ♦s ♣♦♥t♦s P = (x, y) ❡ Q = (x′
, y) tê♠ ❛ ♠❡s♠❛ ♦r❞❡♥❛❞❛ y ❡♥tã♦ ❛ ❞✐stâ♥❝✐❛ d(P, Q) ❡♥tr❡ ❡❧❡s é ✐❣✉❛❧ à ❞✐stâ♥❝✐❛ |x′
−x| ❡♥tr❡ s✉❛s ♣r♦❥❡çõ❡s s♦❜r❡ ♦ ❡✐①♦ OX✳
❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ P = (x, y) ❡ Q′
= (x, y′
) tê♠ ❛ ♠❡s♠❛ ❛❜❝✐ss❛ x ❡♥tã♦ d(P, Q′
) =
|y′
−y| ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛s ♣r♦❥❡çõ❡s ❞❡P ❡ Qs♦❜r❡ ♦ ❡✐①♦ OY
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✽
❋✐❣✉r❛ ✷✳✷✿ ❉✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦s
❝♦♥s✐❞❡r❛♥❞♦ ♦ ♣♦♥t♦ S = (u, y)✱ ✈❡♠♦s q✉❡ P QS é ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❝✉❥❛
❤✐♣♦t❡♥✉s❛ é P Q✳ ❈♦♠♦ P ❡ S tê♠ ❛ ♠❡s♠❛ ♦r❞❡♥❛❞❛✱ ❡♥q✉❛♥t♦S ❡ Qtê♠ ❛ ♠❡s♠❛
❛❜❝✐ss❛✱ s❡❣✉❡✲s❡ q✉❡
d(P, Q) =|x−u| e d(S, Q) =|y−v|
P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
d(P, Q)2 =d(P, S)2+d(S, Q)2
P♦rt❛♥t♦✱
d(P, Q)2 = (x−u)2+ (y−v)2
❧♦❣♦
d(P, Q) =p(x−u)2+ (y−v)2
❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ P = (x, y)à ♦r✐❣❡♠ O = (0,0) é
d(O, P) =px2+y2
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✾
❋✐❣✉r❛ ✷✳✸✿ P❧❛♥♦ ❈❛rt❡s✐❛♥♦
❊①❡♠♣❧♦ ✷✳✶✳✶✳ ✳ ❙❡ ♦ ❝❡♥tr♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C é ♦ ♣♦♥t♦ A = (a, b) ❡ ♦ r❛✐♦ é ♦ ♥ú♠❡r♦ r❡❛❧ r > 0 ❡♥tã♦✱ ♣♦r ❞❡✜♥✐çã♦✱ ✉♠ ♣♦♥t♦ P = (x, y) ♣❡rt❡♥❝❡ ❛ C s❡✱ ❡
s♦♠❡♥t❡ s❡✱ d(A, P) = r✳
P❡❧❛ ❢ór♠✉❧❛ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✱ ✈❡♠♦s q✉❡
C = (x, y); (x−a)2+ (y−b)2 =r2
❉✐③✲s❡ ❡♥tã♦ q✉❡✱
(x−a)2+ (y−b)2 =r2
é ❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ♥♦ ♣♦♥t♦ A= (a, b) ❡ ♦ r❛✐♦r✳
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✵
✷✳✷ ❊▲■P❙❊
❯♠❛ ❡❧✐♣s❡ é ♦❜t✐❞❛ ❝♦♠♦ ✉♠❛ s❡❝çã♦ ❝ô♥✐❝❛✱ s❡ ♦ ♣❧❛♥♦ s❡❝❛♥t❡ ♥ã♦ ❢♦r ♣❛r❛❧❡❧♦ ❛ ♥❡♥❤✉♠❛ ❣❡r❛tr✐③ ❡✱ ♥❡st❡ ❝❛s♦✱ ♦ ♣❧❛♥♦ ✐♥t❡r❝❡♣t❛ t♦❞❛s ❛s ❣❡r❛tr✐③❡s ❝♦♠♦ ♥❛ ✜❣✉r❛ ✷✳✺✳ ❯♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❛ ❡❧✐♣s❡ é ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❝♦♥❢♦r♠❡ ♠♦str❛ ♥❛ ✜❣✉r❛ ✷✳✺ à ❞✐r❡✐t❛✱ ❢♦r♠❛❞❛ q✉❛♥❞♦ ♦ ♣❧❛♥♦ s❡❝❛♥t❡ q✉❡ ✐♥t❡r❝❡♣t❛ t♦❞❛s ❛s ❣❡r❛tr✐③❡s t❛♠❜é♠ ❢♦r ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ ❞♦ ❝♦♥❡✳ ❱❛♠♦s ❞❡✜♥✐r ❛❣♦r❛ ❛ ❡❧✐♣s❡ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ♥✉♠ ♣❧❛♥♦✳
❋✐❣✉r❛ ✷✳✺✿ ❊❧✐♣s❡
❊❧✐♣s❡ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ❝✉❥❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛ ❛ ❞♦✐s ♣♦♥t♦s ✜①♦s é ❝♦♥st❛♥t❡✳ ❖s ♣♦♥t♦s ✜①♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❢♦❝♦s✳
❙❡❥❛2c❛ ❞✐stâ♥❝✐❛ ♥ã♦ ♦r✐❡♥t❛❞❛ ❡♥tr❡ ♦s ❢♦❝♦s✱ ♦♥❞❡ c >0✳ P❛r❛ ♦❜t❡r ❛ ❡q✉❛çã♦
❞❡ ✉♠❛ ❡❧✐♣s❡ ❡s❝♦❧❤❡♠♦s ♦ ❡✐①♦x❝♦♠♦ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ❢♦❝♦sF ❡F′❡ ❡s❝♦❧❤❡♠♦s
❛ ♦r✐❣❡♠ ❝♦♠♦ s❡♥❞♦ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ F F′✳
❖s ❢♦❝♦s F ❡ F′ tê♠ ❝♦♦r❞❡♥❛❞❛s
(c,0)❡ (c,0)✱ r❡s♣❡❝t✐✈❛♠❡♥t♦❡✳ ❙❡❥❛ 2a ❛ s♦♠❛
❝♦♥st❛♥t❡ ♠❡♥❝✐♦♥❛❞❛ ♥❛ ❞❡✜♥✐çã♦✳ ❊♥tã♦✱ a > c ❡ ♦ ♣♦♥t♦ P(x, y) ❞❛ ❋✐❣✉r❛ ✸ s❡rá ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ ❡❧✐♣s❡ s❡ ❡ s♦♠❡♥t❡ s❡ |F P| ✰|F′
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✶
❈♦♠♦
|F P|=p(x−c)2 +y2 e |F′
P|=p(x+c)2+y2,
P ❡stá s♦❜r❡ ❛ ❡❧✐♣s❡ s❡ ❡ s♦♠❡♥t❡ s❡
p
(x−c)2+y2+p(x+c)2+y2 = 2a
❱❛♠♦s s✐♠♣❧✐✜❝❛r ❡ss❛ ❡q✉❛çã♦ ❡s❝r❡✈❡♥❞♦✲❛ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ✉♠ r❛❞✐❝❛❧ ✜q✉❡ à ❡sq✉❡r❞❛ ❡ ♦✉tr♦ à ❞✐r❡✐t❛ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❡❧❡✈❛r❡♠♦s ❛♦ q✉❛❞r❛❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s✳ ❆ss✐♠ ♦❜t❡♠♦s
p
(x−c)2+y2 = 2a−p(x+c)2+y2
(x−c)2+y2 = 4a2 −4ap(x+c)2+y2+ (x+c)2+y2
x2−2cx+c2+y2 = 4a2 −4ap(x+c)2+y2+x2+ 2cx+c2+y2
4ap(x+c)2+y2 = 4a2 + 4ax p
(x+c)2+y2 = a+ c
ax x2(1− c
2
a2) +y
2 = a2 −c2
(a2−c2)x2+a2y2 = a2(a2−c2)
x2 a2 +
y2
(a2−c2) = 1
❈♦♠♦a > c✱a2−c2 >0 ❡ ♣♦❞❡♠♦s ❢❛③❡r b2 =a2−c2
❙✉❜st✐t✉✐♥❞♦ ❡ss❛ ❡q✉❛çã♦ ❡♠
x2
a2 +
y2
(a2 −c2) = 1
t❡♠♦s
x2
a2 +
y2
(b2) = 1
❆ss✐♠ ♠♦str❛♠♦s q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s (x, y) ❞❡ q✉❛❧q✉❡r ♣♦♥t♦ P s♦❜r❡ ❛ ❡❧✐♣s❡
s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦
x2 a2 +
y2 b2 = 1
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✷
✷✳✸ ◆Ú▼❊❘❖❙ ❈❖▼P▲❊❳❖❙
✷✳✸✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛
❊♠ 1545✱ ❏❡rô♥✐♠♦ ❈❛r❞❛♥♦ (1501−1576)✱ ❡♠ s❡✉ ❧✐✈r♦ ✧❆rs ▼❛❣♥❛✧✭❆ ●r❛♥❞❡ ❆rt❡✮✱ ♠♦str♦✉ ♦ ♠ét♦❞♦ ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ t❡r❝❡✐r♦ ❣r❛✉ q✉❡ é ❤♦❥❡ ❝❤❛♠❛❞♦ ❞❡ ❋ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✳ ❇♦♠❜❡❧❧✐(1526−1572)✱ ❞✐s❝í♣✉❧♦ ❞❡ ❈❛r❞❛♥♦✱ ❡♠ s✉❛ ✧➪❧❣❡❜r❛✧✱ ❛♣❧✐❝♦✉ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦ à ❡q✉❛çã♦ x3−15x−4 = 0 ♦❜t❡♥❞♦
x=p3
2 +√−121 +p3
2−√−121
❊♠❜♦r❛ ♥ã♦ s❡ s❡♥t✐ss❡ ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ✈♦♥t❛❞❡ ❡♠ r❡❧❛çã♦ às r❛í③❡s q✉❛❞r❛❞❛s ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✱ ❇♦♠❜❡❧❧✐ ♦♣❡r❛✈❛ ❧✐✈r❡♠❡♥t❡ ❝♦♠ ❡❧❛s✱ ❛♣❧✐❝❛♥❞♦✕❧❤❡s ❛s r❡❣r❛s ✉s✉❛✐s ❞❛ ➪❧❣❡❜r❛✳
◆♦ ❝❛s♦✱ ❇♦♠❜❡❧❧✐ ♠♦str♦✉ q✉❡
(2 +√−1)3 = 23+ 3.22√−1 + 3.2(√−1)2 + (√−1)3 = 8 + 12√−1−6−√−1
= 2 + 11√−1
= 2 +√−121.
▲♦❣♦✱
3
q
2 +√−121 = 2 +√−1
❡✱ ❛♥❛❧♦❣❛♠❡♥t❡✱
3
q
2 +√−121 = 2−√−1
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✸
✷✳✸✳✷ ❆ ❋♦r♠❛ ❆❧❣é❜r✐❝❛
❯♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ é ✉♠ ♥ú♠❡r♦ ❞❛ ❢♦r♠❛ x+yi✱ ❝♦♠ x ❡ y r❡❛✐s ❡ i = √−1✳ ❋✐①❛♥❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ ♣❧❛♥♦✱ ♦ ❝♦♠♣❧❡①♦ z = x+yi é r❡♣r❡s❡♥t❛❞♦
♣❡✇❧♦ ♣♦♥t♦ P(x, y)✳ ❖ ♣♦♥t♦ P é ❝❤❛♠❛❞♦ ❞❡ ✐♠❛❣❡♠ ❞♦ ❝♦♠♣❧❡①♦ z✳ ❈♦♠♦ ❛
❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ❝♦♠♣❧❡①♦s ❡ s✉❛s ✐♠❛❣❡♥s é ✉♠✕❛✕✉♠✱ ❢r❡q✉❡♥t❡♠❡♥t❡ ✐❞❡♥✲ t✐✜❝❛r❡♠♦s ♦s ❝♦♠♣❧❡①♦s ❛ s✉❛s ✐♠❛❣❡♥s ❡s❝r❡✈❡♥❞♦ (x, y) = x+yi ❖ ♣❧❛♥♦ ♥♦ q✉❛❧
r❡♣r❡s❡♥t❛♠♦s ♦s ❝♦♠♣❧❡①♦ é ❝❤❛♠❛❞♦ ❞❡ ♣❧❛♥♦ ❞❡ ❆r❣❛♥❞✕●❛✉ss✳
❖s ♥ú♠❡r♦s r❡♣r❡s❡♥t❛❞♦s ♥♦ ❡✐①♦ ❞♦s x sã♦ ❞❛ ❢♦r♠❛ (x,0) = x+ 0i✱ ✐st♦ é✱ sã♦
♥ú♠❡r♦s r❡❛✐s✳ P♦r ❡ss❡ ♠♦t✐✈♦✱ ♦ ❡✐①♦ ❞♦s x é ❝❤❛♠❞♦ ❞❡ ❡✐①♦ r❡❛❧✳
❖s ♥ú♠❡r♦s r❡♣r❡s❡♥t❛❞♦s ♥♦ ❡✐①♦ ❞♦s y sã♦ ❞❛ ❢♦r♠❛ (0, y) = 0 +yi✱ ✐st♦ é✱ sã♦
♥ú♠❡r♦s r❡❛✐s✳ ❊ss❡s ❝♦♠♣❧❡①♦s sã♦ ❝❤❛♠❞♦s ❞❡ ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s ♣✉r♦s✳
❆s ❝♦♦r❞❡♥❛❞❛sx, y ❞♦ ❝♦♠♣❧❡①♦z =x+yisã♦ ❝❤❛♠❛❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ ❞❡ ♣❛rt❡
r❡❛❧ ❡ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ z0✳ ❊s❝r❡✈❡✲s❡ Re(z) =x ❡ im(z) =y✳
❖♣❡r❛çõ❡s ♥❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛
❆❞✐çã♦
❙❡❥❛ z1 =x1 +y1i ❡z2 =x2+y2i✱ ❞❡✜♥✐♠♦s ❛❞✐çã♦ ❞❡z1+z2 ♣♦r✿
z1+z2 = (x1+y1i) + (x2+y2i)
= (x1+x2) + (y1+y2)i
▼✉❧t✐♣❧✐❝❛çã♦
❉❛❞♦s z1 =x1 +y1i ❡z2 =x2+y2i✱ ❞❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡z1·z2 ♣♦r✿
z1·z2 = (x1+y1i)·(x2+y2i)
= x1·x2+x1·y2i+x2·y1i+x2·y2i2
= (x1·x2−x2·y2) + (x1·y2+x2·y1)i
❖ ❝♦♥❥✉❣❛❞♦ ❞♦ ❝♦♠♣❧❡①♦z = x+yi✱ x ❡ y r❡❛✐s✱ é ♦ ❝♦♠♣❧❡①♦z = x−yi✳ ◆♦t❡
q✉❡ ♦ ♣r♦❞✉t♦
z.z = (x+yi).(x−yi) =x2−y2i2 =x2+y2
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✹
❉✐✈✐sã♦
❉❛❞♦s z1 =x1 +y1i ❡z2 =x2+y2i✱ ❞❡✜♥✐♠♦s ❛ ❞✐✈✐sã♦ ❞❡ z1 ♣♦rz2 ❝♦♠♦✿
z1
z2 =
x1+y1i
x2+y2i
= x1+y1i
x2+y2i ·
x2−y2i x2−y2i
= (x1 ·x2−x2·y2) + (−x1 ·y2+x2·y1)i
x2 2 +y22
✷✳✸✳✸ ❆ ❋♦r♠❛ ❚r✐❣♦♥♦♠étr✐❝❛
❙✉♣♦♥❤❛♠♦s ✜①❛❞♦ ✉♠ s✐st❡♠ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ ♣❧❛♥♦✳
❱❛♠♦s ❛❣♦r❛ r❡♣r❡s❡♥t❛r ❝❛❞❛ ❝♦♠♣❧❡①♦z =x+yi♥ã♦ ♠❛✐s ♣❡❧♦ ♣♦♥t♦P = (x, y)✱
♠❛s s✐♠ ♣❡❧♦ ✈❡t♦r −→OP(x, y)✳
❋✐❣✉r❛ ✷✳✻✿ ❋♦r♠❛ ❚r✐❣♦♥♦♠étr✐❝❛
❖ ♠ó❞✉❧♦ ❞❡ ✉♠ ❝♦♠♣❧❡①♦z =x+yié ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ♠ó❞✉❧♦ ❞♦ ✈❡t♦r q✉❡
r❡♣r❡s❡♥t❛✱ ♦✉ s❡❥❛✱ é ♦ ✈❛❧♦r ❞❡ r ❞❛ ❞✐stâ♥❝✐❛ ❞❡ ✉s❛ ✐♠❛❣❡♠ P à ♦r✐❣❡♠✳ P♦rt❛♥t♦✱
|z|=r=px2+y2
❯♠ ❛r❣✉♠❡♥t♦ ❞❡ ✉♠ ❝♦♠♣❧❡①♦ z 6= 0✱ z = x+yi✱ é✱ ♣♦r ❞❡✜♥✐çã♦✱ q✉❛❧q✉❡r ❞♦s
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✺
q✉❡ t♦❞♦ ❝♦♠♣❧❡①♦ ♥ã♦✕♥✉❧♦ t❡♠ ✉♠❛ ✐♥❥✐♥✐❞❛❞❡ ❞❡ ❛r❣✉♠❡♥t♦s✱ ❞♦✐s q✉❛✐sq✉❡r ❞❡❧❡s ❞✐❢❡r✐♥❞♦ ❡♥tr❡ s✐ ♣♦r ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 2π✳ ❖ ❛r❣✉♠❡♥t♦ q✉❡ ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦
(−π, π] é ❝❤❛♠❞♦ ❞❡ ❛r❣✉♠❡♥t♦ ♣r✐♥❝✐♣❛❧ ❡ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ❆r❣ z✳
❙❡θ é ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ z =x+yi ❡♥tã♦ x= r.cosθ ❡ y = r.senθ✱ ♦ q✉❡ ♣❡r♠✐t❡
❡s❝r❡✈❡r z = x+ yi = r.cosθ + r.senθ = r(cosθ +senθ)✱ q✉❡ é ❛ ❝❤❛♠❛❞❛ ❢♦r♠❛
tr✐❣♦♥♦♠étr✐❝❛ ♦✉ ♣♦❧❛r ❞♦ ❝♦♠♣❧❡①♦ Z✳
❊①❡♠♣❧♦ ✷✳✸✳✶✳ ❊s❝r❡✈❛ ❛ ❢♦r♠❛ tr✐❣♦♥♦♠étr✐❝❛ ❡ r❡♣r❡s❡♥t❡ ❣r❛✜❝❛♠❡♥t❡ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z=−1 +√3i✳
❘❡s♦❧✉çã♦✿ ❱❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❞❡t❡r♠✐♥❛r ♦ ♠ó❞✉❧♦ ❞❡ z✱
|z|=r=px2+y2 = q
(−1)2+ (√3)2 = 2
❆❧é♠ ❞✐ss♦✱
cosθ = x
r =−
1
2 e senθ =
y r =
√
3 2
▲♦❣♦✱ ✉♠ ❞♦s ✈❛❧♦r❡s ♣♦ssí✈❡✐s ♣❛r❛ θ é 2π
3 ❡ ❛ ❢♦r♠❛ tr✐❣♦♥♦♠étr✐❝❛ ❞❡ z é
z= 2
cos2π
3 +isen 2π
3
✷✳✹ ❊◗❯❆➬Õ❊❙ P❖▲■◆❖▼■❆■❙
✷✳✹✳✶ P♦❧✐♥ô♠✐♦s ❈♦♠♣❧❡①♦s
❯♠❛ ❢✉♥çã♦ p:C → Cé ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ q✉❛♥❞♦ ❡①✐st❡♠ ♥ú♠❡r♦s
❝♦♠♣❧❡①♦s a0, a1, ..., an t❛✐s q✉❡
p(x) =anxn+an−1x
n−1
+an−2x
n−2
+· · · +a1x+a0
♣❛r❛ t♦❞♦ x ∈ C✳ ❖s ♥ú♠❡r♦sa0, a1,· · · , an sã♦ ♦s ❝♦❡✜❝✐❡♥t❡r ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧✳
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✻
❋✐❣✉r❛ ✷✳✼✿ ❊①❡♠♣❧♦
❊①❡♠♣❧♦ ✷✳✹✳✶✳ ❙❡❥❛ p :C→ C t❛❧ q✉❡ p(x) =x2+ 1✳ ❆ r❡str✐çã♦ ❞❡ p ♣❛r❛ R é ❛
❢✉♥çã♦ q✉❛❞rát✐❝❛ f :R →R t❛❧ q✉❡ f(x) = x2+ 1✳
❊✈✐❞❡♥t❡♠❡♥t❡✱ f(x) > 0 ♣❛r❛ t♦❞♦ x✱ ♦ q✉❡ ♠♦str❛ q✉❡ f ♥ã♦ t❡♠ r❛í③❡s ❡ q✉❡ p ♥ã♦ t❡♠ r❛í③❡s r❡❛✐s✳ ▼❛s p é ❞❡✜♥✐❞❛ ❡♠ t♦❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❊♠ ♣❛rt✐❝✉❧❛r p(i) = i2+ 1 =−1 + 1 = 0✳ P♦rt❛♥t♦ i é ✉♠❛ r❛✐③ ❝♦♠♣❧❡①❛ ❞❡ p✳
❙♦♠❛s ❡ ♣r♦❞✉t♦s ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❝♦♠♣❧❡①❛s sã♦✱ t❛♠❜é♠✱ ❢✉♥çõ❡s ♣♦❧✐♥♦✲ ♠✐❛✐s ❝♦♠♣❧❡①❛s✳ ◆♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ❛♠❜❛s ❢✉♥çõ❡s t❡♠ t♦❞♦s ♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✱ ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ t❛♠❜é♠ tê♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✳ ◆♦t❡ q✉❡✱ s❡p❡qsã♦ ❢✉♥çõ❡s
♣♦❧✐♥♦♠✐❛✐s✱ ❡♥tã♦ ♦ ❣r❛✉ ❞❡ p+q é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛♦ ♠❛✐♦r ❡♥tr❡ ♦s ❣r❛✉s
❞❡ p ❡ q✱ ❡♥q✉❛♥t♦ ♦ ❣r❛✉ ❞❡ pq é ❛ s♦♠❛ ❞♦s ❣r❛✉s ❞❡ p ❡ q✳ ◗✉❛♥❞♦ ✉♠❛ ❢✉♥çã♦
♣♦❧✐♥♦♠✐❛❧ p ♣♦❞❡ s❡r ❡①♣r❡ss❛ ❝♦♠♦ ♦ ♣r♦❞✉t♦ p =qr ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s q ❡ r✱
❞✐①③❡♠♦s q✉❡ pé ❞✐✈✐sí✈❡❧ ♣♦r q ❡r✳
❊①❡♠♣❧♦ ✷✳✹✳✷✳ ❆ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ P(x) = xn−αn é ❞✐✈✐sí✈❡❧ ♣♦r x−α✱ ♦♥❞❡ α
✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ q✉❛❧q✉❡r ❇❛st❛ ✈❡r✐✜❝❛r q✉❡
xn−αn = (x−α)(xn−1
+αxn−2
+α2xn−3
+...+αn−2
x+αn−1
)
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✼
❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ p(α) = 0
p(x) = p(x)−p(α) = (anxn+an−1x
n−1
+· · ·+a1x+a0)−(anαn+an−1α
n−1
+· · ·+a1α+a0)
= an(xn−αn) +an−1(x
n−1
−αn−1
) +· · ·+a1(x−α)
❉♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r t❡♠♦s q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ♣❛r❝❡❧❛s ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ é ❞✐✈✐sí✈❡❧ ♣♦r x−α✳ ▲♦❣♦ p(x) é ❞✐✈✐sí✈❡❧ ♣♦r (x−α)❀ ✐st♦ é✱ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ q t❛❧ q✉❡ p(x) =q(x)(x−α)✳
❉❡ ♠♦❞♦ ♠❛✐s ❣❡r❛❧✱ s❡ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s α1, α2,· · ·, αk✱ sã♦ r❛í③❡s ❞✐st✐♥t❛s
❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ p ❞❡ ❣r❛✉ n✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ q ❞❡ ❣r❛✉ n−k t❛❧ q✉❡
p(x) = (x−α1)(x−α2) · · · (x−αk)q(x)
P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉ n ♣♦❞❡ t❡r✱ ♥♦
♠á①✐♠♦✱ n r❛í③❡s✳
◆♦t❡ t❛♠❜é♠ q✉❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ♥ã♦ ♣♦❞❡ s❡r ♥✉❧❛ ♣❛r❛ t♦❞♦ ✈❛❧♦r ❞❡
x✱ ❛ ♠❡♥♦s q✉❡ t♦❞♦s ♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s s❡❥❛♠ ♥✉❧♦s✳ ■st♦ é p ❞❡✈❡ s❡r ❞❛ ❢♦r♠❛ p(x) = 0 + 0x+ 0x2+· · ·✳
❉❡ ❢❛t♦✱ s❡ ❛❧❣✉♠ ❝♦❡✜❝✐❡♥t❡ ❞❡ p ❢♦ss❡ ♥ã♦✲♥✉❧♦✱ ❡♥tã♦ p t❡r✐❛ ✉♠ ❝♦❡✜❝✐❡♥t❡ an ♥ã♦✲♥✉❧♦ ❞❡ í♥❞✐❝❡ ♠á①✐♠♦❀ ♦✉ s❡❥❛✱ t❡r✐❛ ❣r❛✉n✳ P♦rt❛♥t♦✱pt❡r✐❛ ♥♦ ♠á①✐♠♦nr❛í③❡s✱
♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ ps❡ ❛♥✉❧❛r ♣❛r❛ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ x✳
❆ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧pt❛❧ q✉❡p(x) = 0♣❛r❛ t♦❞♦xé ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧
✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳ ❖❜s❡r✈❡ q✉❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♥♦ss❛ ❞❡✜♥✐çã♦✱ p ♥ã♦ ♣♦ss✉✐ ❣r❛✉✱
♣♦r ♥ã♦ ♣♦ss✉✐r ♥❡♥❤✉♠ ❝♦❡✜❝✐❡♥t❡ ♥ã♦ ♥✉❧♦✳
❚♦♠❡♠♦s ❛❣♦r❛ ❞✉❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s p ❡ q✳ P❛r❛ q✉❡ p ❡ q s❡❥❛♠ ✐❣✉❛✐s✱ ✐st♦
é✱ s❡❥❛♠ t❛✐s q✉❡ p(x) = q(x) ♣❛r❛ t♦❞♦ x ∈ C✱ s✉❛ ❞✐❢❡r❡♥ç❛ p−q t❡♠ q✉❡ s❡r
✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳ ▼❛s✱ ❝♦♠♦ ✈✐♠♦s✱ ✐ss♦ ♦❝♦rr❡ s♦♠❡♥t❡ s❡ t♦❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡
p−q sã♦ ♥✉❧♦s❀ ♣♦rt❛♥t♦✱ ❞✉❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s p ❡ q sã♦ ✐❣✉❛✐s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p ❡ q tê♠ ❝♦❡✜❝✐❡♥t❡s r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐❣✉❛✐s✳
❈❤❛♠❛♠♦s ❞❡ ♣♦❧✐♥ô♠✐♦ ❝♦♠♣❧❡①♦ ❛ ✉♠❛ ❡①♣r❡ssã♦ ❢♦r♠❛❧ ❞♦ t✐♣♦
p(X) =anXn+an−1X
n−1
+· · ·+a1X+a0,
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✽
✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠♦ ❛ ❧✐st❛ ♦r❞❡♥❛❞❛(a0, a1,· · · , an)❞❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s ❡ q✉❡ s♦♠❛♠♦s
❡ ♠✉❧t✐♣❧✐❝❛♠♦s ♣♦❧✐♥ô♠✐♦s ❛tr❛✈és ❞❛s r❡❣r❛s ✉s✉❛✐s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠♦♥ô♠✐♦s ❡ ❛❞✐çã♦ ❞❡ ♠♦♥ô♠✐♦s s❡♠❡❧❤❛♥t❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r ❞✐③❡♠♦s q✉❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s sã♦ ✐❣✉❛✐s q✉❛♥❞♦ ♣♦ss✉❡♠ ❡①❛t❛♠❡♥t❡ ♦s ♠❡s♠♦s ❝♦❡✜❝✐❡♥t❡s✳ ❆ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ❝♦♠♣❧❡①♦
p(X) =anXn+an−1X
n−1
+· · ·+a1X+a0,
❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ p:C → C t❛❧ q✉❡
p(X) =anxn+an−1xn −1
+· · ·+a1x+a0.
◆♦t❡ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦❧✐♥ô♠✐♦ ❝♦♥t❡♠♣❧❛ ❛♣❡♥❛s ❛ ❧✐st❛ ❞❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s ❡ ❛ ❢♦r♠❛ ♣❡❧❛ q✉❛❧ ♦s s♦♠❛♠♦s ❡ ♠✉❧t✐♣❧✐❝❛♠♦s❀ q✉❛♥❞♦ ♥♦s r❡❢❡r✐♠♦s à ❢✉♥çã♦ ♣♦✲ ❧✐♥♦♠✐❛❧✱ ♣❛ss❛♠♦s ❛ ❡st❛r ✐♥t❡r❡ss❛❞♦s ♥❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡st❛❜❡❧❡❝✐❞❛ ♣❡❧♦ ✈❛❧♦r q✉❡ ❛ ❢✉♥çã♦ ❛ss✉♠❡ ❡♠ ❝❛❞❛ ♣♦♥t♦✳ ➱ ❝❧❛r♦ q✉❡ ❛ t♦❞♦ ♣♦❧✐♥ô✲ ♠✐♦ ❝♦rr❡s♣♦♥❞❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ✈✐♠♦s q✉❡ ❞✉❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s só sã♦ ✐❣✉❛✐s q✉❡♥❞♦ tê♠ ❛ ♠❡s♠❛ ❧✐st❛ ❞❡ ❝♦❡✜❝✐❡♥t❡s✳ ❆ss✐♠✱ ❛ ✉♠❛ ❢✉çã♦ ♣♦❧✐♥♦♠✐❛❧ t❛♠❜é♠ ❝♦rr❡s♣♦♥❞❡ ✉♠ ú♥✐❝♦ ♣♦❧✐♥ô♠✐♦✳ ❉❡ss❡ ♠♦❞♦✱ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✲✉♥í✈♦❝❛ ❡♥tr❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❡ ♣♦❧✐♥ô♠✐♦s✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t❡✱ ♥♦s r❡❢❡r✐r♠♦s ✐♥❞✐st✐♥t❛♠❡♥t❡ ❛♦ ♣♦❧✐♥ô♠✐♦ p ♣♦r ❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧p✳
❊①❡♠♣❧♦ ✷✳✹✳✸✳ ❱❡r✐✜❝❛r s❡ ♦ ♣♦❧✐♥ô♠✐♦ p(x) = 9x3−3x2−5x−25 é ❞✐✈✐sí✈❡❧ ♣♦r
3x−5
◗✉❡r❡♠♦s ✈❡r✐✜❝❛r s❡ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ q t❛❧ q✉❡ p(x) = (3x−5)q(x).
❖❜s❡r✈❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡✱ ❝❛s♦ ❡①✐st❛ ♦ ♣♦❧✐♥ô♠✐♦ q✱ ❡❧❡ ❞❡✈❡ s❡r ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱
❞❛ ❢♦r♠❛ q(x) = ax2+bx+c✳ ■st♦ é✱ ♣r♦❝✉r❛♠♦s ♥ú♠❡r♦s a✱ b ❡ ct❛✐s q✉❡✿
9x3−3x2−5x−25 = (3x−5)(ax2+bx+c).
❊❢❡t✉❛♥❞♦ ♦ ♣r♦❞✉t♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦✱ ♦❜t❡♠♦s✿
9x3−3x2−5x−25 = 3ax3+ (3b−5a)x2+ (3c−5b)x−5c.
P♦rt❛♥t♦✱ ❞❡✈❡✕s❡ t❡r✿
3a = 9 3b−5a = −3 3c−5b = −5
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✾
◆❛ ✶❛ ❡q✉❛çã♦✱ ♦❜t❡♠♦s a = 3✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛ ✷❛✱ ✈❡♠ b = 4✳ ❋✐♥❛❧♠❡♥t❡✱ ❞❛
✸❛ ✈❡♠ q✉❡ c−5✳ ❈♦♠♦ ❛ ✹❛ ❡q✉❛çã♦ é s❛t✐s❢❡✐t❛ ♣♦r ❡ss❡s ✈❛❧♦r❡s✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛
r❡s♣♦st❛ é ❛✜r♠❛t✐✈❛✳ ■st♦ é✱ p(x) é ❞✐✈✐sí✈❡❧ ♣♦r 3x−5❡ q(x) = 3x2+ 4x+ 5.
✷✳✹✳✷ ❉✐✈✐sã♦ ❞❡ P♦❧✐♥ô♠✐♦s
❙❡ ✉♠ ♣♦❧✐♥ô♠✐♦p ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ♦ ♣r♦❞✉t♦ p=p1p2 ❞❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s p1 ❡ p2✱ ❡♥tã♦ ✉♠ ❝♦♠♣❧❡①♦ α é r❛✐③ ❞❡p s❡✱ ❡ s♦♠❡♥t❡ s❡✱α é ❛ r❛✐③ ❞❡ p1 ♦✉ r❛✐③ ❞❡p2✱ ❥á
q✉❡
p1(x)p2(x) = 0⇒p1(x) = 0 ou p2(x) = 0.
❈♦s♥❞✐❞❡r❡♠♦s✱ ❡♥tã♦✱ ❛ s❡❣✉✐♥t❡ q✉❡stã♦✿ ❞❛❞♦s ♣♦❧✐♥ô♠✐♦sD ❡ d✱ ❝♦♠♦ ✈❡r✐✜❝❛r
s❡ Dé ❞✐✈✐sí✈❡❧ ♣r♦d ❄ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ é r❡♣❡t✐r ♦ q✉❡ ✜③❡♠♦s ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ❡
✈❡r✐✜❝❛r ❛ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❞❡D♣♦rdt❡♥t❛♥❞♦ ❡♥❝♦♥ tr❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ q t❛❧ q✉❡ D=qd✳
❯♠❛ ♦✉tr❛ ❛❧t❡r♥❛t✐✈❛ é ♦❧❤❛r♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛✐s ❣❡r❛❧✱ q✉❡ ♥♦s ❧❡✈❛ ❛ ✉♠ ♣r♦❝❡ss♦ ❛❧❣♦rít♠✐❝♦ ❞❡ ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❛tr❛✈és ❞♦ q✉❛❧ ♣♦❞❡♠♦s ♦❜t❡r q ♦✉ ❞❡♠♦♥str❛r q✉❡ D♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r d✳ ■ss♦ é ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣♦ ❛♦ q✉❡
♦❝♦rr❡ ❝♦♠ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✿ ❛ ❢♦r♠ ♠❛✐s ♣rát✐❝❛ ❞❡ ✈❡r✐✜❝❛r s❡ ♦ ♥ú♠❡r♦ ✷✹✽✳✾✽✼ é ❞✐✈✐sí✈❡❧ ♣r♦ ✷✶✶ ❝♦♥s✐st❡ ❡♠ ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❡ ✈❡r✐✜❝❛r s❡ ♦ r❡st♦ é ♦✉ ♥ã♦ ✐❣✉❛❧ ❛ ③❡r♦✳
❱❛♠♦s r❡❝♦r❞❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐✈✐sã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❉❛❞♦ ✉♠ ✐♥t❡✐r♦ ❞✐✈✐✲ ❞❡♥❞♦ D ❡ ✉♠ ✐♥t❡✐r♦ ❞✐✈✐s♦r d 6= 0✱ ❞✐✈✐❞✐r D ♣♦r d ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥t❛r ✐♥t❡✐r♦s q ❡ r✱ ♦♥❞❡ 0 ≤ r ≤ |d| −1✱ ❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ q✉❛❝✐❡♥t❡ ❡ r❡st♦ ❞❛ ❞✐✈✐sã♦✱
q✉❡ ❝✉♠♣r❛♠ D=dq+r✳
❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❞✐✈✐❞✐r ✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠♣❧❡①♦ d ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠♣❧❡①♦ d ♥ã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❝♦♥s✐st❡ ❡♠ ♦❜t❡r ♣♦❧✐♥ô♠✐♦s ❝♦♠♣❧❡①♦s q ❡ r✱ ❝❤❛♠❛❞♦s
r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ q✉♦❝✐❡♥t❡ ❡ r❡st♦ ❞❛ ❞✐✈✐sã♦✱ q✉❡ ❝✉♠♣r❛♠✿
grau(r)< grau(d) e D=dq+r. ✭✷✳✶✮
❚❡♦r❡♠❛ ✷✳✹✳✷✳ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦D♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ d✱ ♥ã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✱ ❡①✐st❡♠ ❡ sã♦ ú♥✐❝♦s✳
❉❡♠♦♥str❛çã♦✳ ❈♦♠❡❝❡♠♦s ❝♦♠ ❛ ✉♥✐❝✐❞❛❞❡✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛♠ ❞♦✐s ♣❛r❡s ❞❡ ♣♦❧✐♥ô♠✐♦s (q1, r1) ❡ (q2, r2) s❛t✐s❢❛③❡♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐sã♦ ❞❡ D ♣♦rd✳ ■st♦ é✿
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✸✵
D=dq2+r2.
❚❡♠♦s
dq1+r1 =dq2 +r2
❡
d(q1−q2) = r2−r1
◆♦t❡ q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ t❡♠ ❣r❛✉ ♠❡♥♦r q✉❡ ♦ ❣r❛✉ ❞❡ d✱ ♣♦r s❡r ❛
❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ ♠❡♥♦r ❞♦ q✉❡ ♦ ❣r❛✉ ❞❡ d✳ ❏á ♦ ♣♦❧✐♥ô♠✐♦ ❞❛
❡sq✉❡r❞❛ t❡♠ ❣r❛✉ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛♦ ❞❡ d✱ ❛ ♠❡♥♦s q✉❡ (q1−q2) s❡❥❛ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✳ ▲♦❣♦✱ ❛ ✐❞❡♥t✐❞❛❞❡ ♦❝♦rr❡ s♦♠❡♥t❡ q✉❛♥❞♦ ♦s ♣♦❧✐♥ô♠✐♦s ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s sã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦s✳ P♦rt❛♥t♦✱ t❡♠♦s✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ q1 =q2 ❡ r1 =r2✳
P❛r❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛✱ ❡♠♣r❡❣❛r❡♠♦s ✉♠ ♣r♦❝❡ss♦ ❛❧❣♦rít♠✐❝♦ ❛tr❛✈és ❞♦ q✉❛❧ r❡❞✉③✐r❡♠♦s s✉❝❡ss✐✈❛♠❡♥t❡ ♦ ❣r❛✉ ❞♦ ❞✐✈✐❞❡♥❞♦ ❛té q✉❡ ❡❧❡ s❡ t♦r♥❡ ♠❡♥♦r q✉❡ ♦ ❞♦ ❞✐✈✐s♦r ❡ ❛ ❞✐✈✐sã♦ s❡ t♦r♥❡ ♠❡♥♦r q✉❡ ♦ ❞♦ ❞✐✈✐s♦r ❡ ❛ ❞✐✈✐sã♦ s❡ t♦r♥❡ ✐♠❡❞✐❛t❛✳ ◆♦t❡ q✉❡✱ s❡ Dt❡♠ ❣r❛✉ ♠❡♥♦r q✉❡d✱ ❡♥tã♦ ❝❡rt❛♠❡♥t❡ D♣♦❞❡ s❡r ❞✐✈✐❞✐❞♦ ♣♦r d✱ ❥á
q✉❡ q = 0 ❡ r = D ❝✉♠♣r❡♠ ❛s ❝♦♥❞✐çõ❡s ❡♠ ✷✳✶✳ ❙✉♣♦♥❤❛♠♦s✱ ❡♥tã♦✱ q✉❡ D t❡♥❤❛
❣r❛✉ n ❡ d t❡♥❤❛ ❣r❛✉ m✳ ❙❡ m > n✱ ♥ã♦ ❤á ♥❛❞❛ ❛ ❢❛③❡r✿ ♦ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ é q = 0 ❡ ♦ r❡st♦ ér=D✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦s t❡r♠♦sanxn❡ bmxm✱ q✉❡ sã♦ ♦s t❡r♠♦s ❞❡ ♠❛✐s ❛❧t♦ ❣r❛✉ ❡♠ D ❡ d✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ r1 ♦ ♣♦❧✐♥ô♠✐♦ ❞❡✜♥✐❞♦
♣♦r
r1(x) =D(x)− an
bm
xn−md(x).
◆♦t❡ q✉❡ r1 é ♦❜t✐❞♦ s✉❜tr❛✐♥❞♦ ❞❡ D ♦ r❡s✉❧t❛❞♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ d ♣❡❧♦ q✉♦✲
❝✐❡♥t❡ ❞♦s t❡r♠♦s ❞❡ ♠❛✐s ❛❧t♦ ❣r❛✉ ❞❡ D ❡ d❀ r1 é ❝❤❛♠❛❞♦ ❞❡ ♣r✐♠❡✐r♦ r❡st♦ ♣❛r❝✐❛❧
♥♦ ♣♦r❝❡ss♦ ❞❡ ❞✐✈✐sã♦✳ ❖❜s❡r✈❡ q✉❡
an
bm
xn−md (x).
é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ n ❝✉❥♦ t❡r♠♦s ❞❡ ♠❛✐s ❛❧t♦ ❣r❛✉ é ✐❣✉❛❧ ❛♦ t❡r♠♦ ❞❡ ♠❛✐s ❛❧t♦
❣r❛✉ anxn ❞❡ D✳ ▲♦❣♦ r1 t❡♠ ❣r❛✉ ♥♦ ♠á①✐♠♦s ✐❣✉❛❧ ❛ n−1✳ ◆ã♦ s❛❜❡♠♦s ❛✐♥❞❛ s❡
r1 ♣♦❞❡ s❡r ❞✐✈✐❞✐❞♦ ♣♦r d❀ ✐st♦ é✱ s❡ ❡①✐st❡♠ ♣♦❧✐♥ô♠✐♦sq1 ❡ r ❝♦♠ grau(r)< grau(d)
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✸✶
❞✐✈✐❞✐❞♦ ♣♦r d✱ ❥á q✉❡ t❡r❡♠♦s
D(x) = an
bm
xn−m
d(x) +r1(x)
= an
bm
xn−md
(x) +q1(x)d(x) +r(x)
=
a
n
bm
xn−m
+q1(x)
d(x) +r(x)
✐st♦ é✱ ♦ r❡st♦ é ♦ ♠❡s♠♦ q✉❡ ♥❛ ❞✐✈✐sã♦ ❞❡ r1 ♣♦r d✱ ❡♥q✉❛♥t♦ ♦ q✉♦❝✐❡♥t❡ é ♦❜t✐❞♦
s♦♠❛♥❞♦ ❛♦ ♣♦❧✐♥ô♠✐♦ ❞❡ q✱ ❝✉❥♦ ❣r❛✉ é ♥♦ ♠á①✐♠♦ n−m−1✱ ♦ t❡r♠♦ an
bm
xn−md(x).
❡st❛ ❢♦r♠❛✱ r❡❞✉③✐♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞✐✈✐❞✐r D ♣♦r d ❛♦ ❞❡ ❞✐✈✐❞✐r r1 ♣♦r d✱ ♦♥❞❡ r1
t❡♠ ❣r❛✉ ♠❛✐s ❜❛✐①♦✳ ▼❛s ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ♠❡s♠♦ ♣r♦❝❡ss♦ ❛ r1✱ ♦❜t❡♥❞♦ ✉♠ ♥♦✈♦
r❡st♦ ♣❛r❝✐❛❧ r2 ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ s❡♠♣r❡ ♦❜t❡♥❞♦ ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ ✐♥❢❡r✐♦r ❛♦
❞♦ ❛♥t❡r✐♦r✳ ❆♣ós ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣❛ss♦s✱ ♦❜t❡r❡♠♦s ✉♠ r❡st♦ ♣❛r❝✐❛ rk ❞❡ ❣r❛✉ ♠❡♥♦r q✉❡ m✱ ♣❛r❛ ♦ q✉❛❧ ❛ ❞✐✈✐sã♦ é ♣♦ssí✈❡❧ ❡ ✐♠❡❞✐❛t❛✿ ♦ q✉♦❝✐❡♥t❡ é qk = 0 ❡ ♦ r❡st♦ é r = rk✳ ❘❡t♦r♥❛♥❞♦ s♦❜r❡ ♥♦ss♦s ♣❛ss♦s✱ ❝♦♥❝❧❧✉í♠♦s q✉❡ ❝❛❞❛ r❡st♦ ♣❛r❝✐❛❧
♣♦❞❡ s❡r ❞✐✈✐❞✐❞♦ ♣♦r d✳ ❖ r❡st♦ ❞❛ ❞✐✈✐sã♦ ♦r✐❣✐♥❛❧ é ✐❣✉❛❧ ❛♦ ú❧t✐♠♦ r❡st♦ ♣❛r❝✐❛❧ rk ❡ ♦ q✉♦❝✐❡♥t❡ é ❢♦r♠❛❞♦ ❝♦❧❡❝✐♦♥❛♥❞♦ ♦s t❡r♠♦s ♦❜t✐❞♦s ❡♠ ❝❛❞❛ ♣❛ss♦✳
❆ss✐♠✱ t❡♠♦s ✉♠❛ ♣r♦✈❛ ❞❡ q✉❡ é ♣♦ssí✈❡❧ ❞✐✈✐❞✐rD ♣♦rd ❡ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ✉♠
♣r♦❝❡ss♦ ♣❛r❛ ❡①❡❝✉t❛r ❛ ❞✐✈✐sã♦ ❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣❛ss♦s✳ ❚♦❞❛ ❛ ❞✐s❝✉ssã♦ ❛❝✐♠❛ é ✈á❧✐❞❛ ♣❛r❛ ♣♦❧✐♥ô♠✐♦s ❝♦♠♣❧❡①♦s✳ ◆♦ ❡♥t❛♥t♦✱ s❡ t♦❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ D
❡ d sã♦ r❡❛✐s✱ ❡♥tã♦ t♦❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❣❡r❞♦s ♥♦ ♣r♦❝❡ss♦ sã♦ ♦❜t✐❞♦s ❛tr❛✈és ❞❡
♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ s♦♠❡♥t❡ ♥ú♠❡r♦s r❡❛✐s ❡ sã♦✱ ♣♦rt❛♥t♦✱ r❛✐s✳ ▲♦❣♦✱ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ r❡❛❧ ♣♦r ♦✉tr♦ sã♦ t❛♠❜❡♠ ♣♦❧✐♥ô♠✐♦s r❡❛✐s✳
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✸✷
❚❡♠♦s D(x) =x4−3x3+ 4x2−6x+ 1✱ ❡d(x) =x−2✳
r1 = D(x)−
1 1
x4−1
d(x)
= x4−3x3 + 4x2−6x+ 1−x3(x−2) = −x3+ 4x2−6x+ 1
r2 = r1(x)−
−1 1
x3−1
d(x)
= −x3+ 4x2−6x+ 1 +x2(x−2) = 2x2 −6x+ 1
r3 = r2(x)−
2 1
x2−1
d(x) = 2x2 −6x+ 1−2x(x−2) = −2x+ 1
r4 = r3(x)−
−2 1
x2−1
d(x) = −2x+ 1 + 2(x−2) = −3
❈♦♠♦r4 t❡♠ ❣r❛✉ ♠❡♥♦r q✉❡ ♦ ❞❡ d(x)✱ ♦ ♣r♦❝❡ss♦ ❡stá t❡r♠✐♥❛❞♦✱ ❝♦♠ q✉♦❝✐❡♥t❡ q(x) = x3−x2+ 2x−2 ❡ ♦ r❡st♦ r(x) = −3.
✷✳✹✳✸ ❉✐✈✐sã♦ ❞❡ ✉♠ P♦❧✐♥ô♠✐♦ ♣♦r
(x
−
α)
❖ ❝❛s♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❡ ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦ é ❛q✉❡❧❡ ❡♠ q✉❡ ♦ ❞✐✈✐s♦r é ❞❛ ❢♦r♠❛ (x−α)✳ ❙❡♠♣r❡ q✉❡ ✉♠ ♥ú♠❡r♦ α é ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠♦ ✉♠❛ r❛✐③ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ p(x)✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ p(x) é ❞✐✈✐sí✈❡❧ ♣♦r (x−α)✳ ❖ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ♦❢❡r❡❝❡ ✉♠ ♦✉tr♦ ♠♦❞♦ ❞❡ ❝❤❡❣❛r à ♠❡s♠❛ ❝♦♥❝❧✉sã♦✳ ❉❡ ❢❛t♦✱ ❛♦ ❞✐✈✐❞✐r ✉♠ ♣♦❧✐♥ô♠✐♦ q✉❛❧q✉❡r ♣♦r (x−α) ♦❜t❡♠♦s ✉♠ q✉♦❝✐❡♥t❡ q(x) ❡ ✉♠ r❡st♦ r(x) = r0✱ s❛t✐s❢❛③❡♥❞♦
p(x) = (x−α)q(x) +r0✳ ❈❛❧❝✉❧❛♥❞♦ ♦ ✈❛❧♦r ♥✉♠ér✐❝♦ ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s ♣❛r❛ x=α✱
♦❜t❡♠♦s p(x) = r0.
❆ss✐♠✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦p(x)♣♦r(x−α)é ♦ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ p(α)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♥❝❧✉í♠♦s q✉❡ ✉♠ ♥ú❡r♦ α é r❛✐③ ❞❡ p(x) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p(x) é ❞✐✈✐sí✈❡❧ ♣♦r x−α.
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✸✸
α ❞❡ p(x)✱ ❛ ❞✐✈✐sã♦ ❞❡ p(x) ♣♦r (x−α) ♣❡r♠✐t❡ ♦❜t❡r ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ♠❡♥♦r ❣r❛✉ ❝✉❥❛s r❛í③❡s sã♦ ❛s ❞❡♠❛✐s r❛í③❡s ❞❡ p(x)✳
❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣♦❧✐♥ô♠✐♦
p(x) =anxn+an−1x
n−1
+an−2x
n−2
+· · ·+a1x+a0
❙❡❥❛♠
q(x) = bn−1x
n−1
+bn−2x
n−2
+· · ·+b1x+b0
❡(x) =r0 ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❛ ❞✐✈✐sã♦ ❞❡p(x)♣♦r(x−α)✳ ❉❡ss❛
❢♦r♠❛✱ t❡♠♦s p(x) = q(x)(x−α) +r0✳ ❉❡s❡♥✈♦❧✈❡♥❞♦ ❛ ❡①♣r❡ssã✐♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦
s❡❣✉♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ x✱ ♦❜t❡♠♦s✿
bn−1x
n+ (b
n−2−αbn−1)x
n−1
+ (bn−3−αbn−2)x
n−2
+ (b0−αb1)x+ (r0−αb0).
■❣✉❛❧❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❛♦s ❞♦s t❡r♠♦s r❡s♣❡❝t✐✈♦s ❞❡p✱ ♦❜t❡♠♦s
bn−1 =an
❡st❛ ❡q✉❛çã♦ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ♦ ❝♦❡✜❝✐❡♥t❡ ❞♦ t❡r♠♦ ❞❡ ♠❛✐s ❛❧t♦ ❣r❛✉ ❞❡ q✳
bn−2−αbn−1 =an−1 ⇒bn−2 =an−1 +αbn−1
❡①♣r✐♠❡ ♦ ♣ró①✐♠♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ q ❡♠ ❢✉♥çã♦ ❞♦ ♦❜t✐❞♦ ♥❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r bn−3−αbn−2 =an−2 ⇒bn−3 =an−2 +αbn−2
r❡♣❡t✐♥❞♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ✈❡③❡s ✈❛♠♦s ❡♥❝♦♥tr❛r ♦ ú❧t✐♠♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ q r0−αb0 =a0 ⇒r0 =a0+αb0.
❈♦♠♦ ♦s ❝á❧❝✉❧♦s ❛❝✐♠❛ ♠♦str❛♠✱ t❡♠♦s ✉♠ ♣r♦❝❡ss♦ r❡❝✉rs✐✈♦ ♣❛r❛ ♦❜t❡r s✉❝❡s✲ s✐✈❛♠❡♥t❡ ♦s t❡r♠♦s ❞❡ q✱ ❛ ♣❛rt✐r ❞♦ t❡r♠♦ ❞❡ ♠❛✐s ❛❧t♦ ❣r❛✉✱ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦✳
❊✈✐❞❡♥t❡♠❡♥t❡✱ ♣♦❞❡rí❛♠♦s t❡r ❝❤❡❣❛❞♦ à ♠❡s♠❛ ❝♦♥❝❧✉sã♦ ❛❝♦♠♣❛♥❤❛♥❞♦✱ ♣❛ss♦✲❛✲ ♣❛ss♦✱ ♦ ❛❧❣♦r✐t♠♦ ❣❡♥ér✐❝♦ ❞❡ ❞✐✈✐sã♦ ❝♦♠ ❞✐✈✐s♦r ❞❛ ❢♦r♠ x−α✳
❖s ❝á❧❝✉❧♦s ❞❡s❝r✐t♦s ❛❝✐♠❛ sã♦ ❢❛❝✐❧♠❡♥t❡ ❡❢❡t✉❛❞♦s q✉❛♥❞♦ ❞✐s♣♦st♦s ♥❛ ❢♦r♠❛ ❛❜❛✐①♦✱ q✉❡ ❝♦♥st✐t✉✐ ♦ ❝❤❛♠❛❞♦ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐✳
an an−1 an−2 · · · a2 a1 a0
