Open Equações polinomiais: soluções algébricas, geométricas e com o auxílio de derivadas

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✿

❙♦❧✉çõ❡s ❛❧❣é❜r✐❝❛s✱ ❣❡♦♠étr✐❝❛s ❡

❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ❞❡r✐✈❛❞❛s

♣♦r

❘♦♥❛❧❞♦ ❞❛ ❙✐❧✈❛ P♦♥t❡s

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

P814e Pontes, Ronaldo da Silva.

Equações polinomiais: soluções algébricas, geométricas e com o auxílio de derivadas / Ronaldo da Silva Pontes.-- João Pessoa, 2013.

89f. : il.

Orientador: Napoleón Caro Tuesta Dissertação (Mestrado) – UFPB/CCEN

1. Matemática. 2. Polinômios. 3. Equações polinomiais.

❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✿

❙♦❧✉çõ❡s ❛❧❣é❜r✐❝❛s✱ ❣❡♦♠étr✐❝❛s ❡

❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ❞❡r✐✈❛❞❛s

♣♦r

❘♦♥❛❧❞♦ ❞❛ ❙✐❧✈❛ P♦♥t❡s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ➪❧❣❡❜r❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ●✐❧❜❡rt♦ ❋❡r♥❛♥❞❡s ❱✐❡✐r❛ ✲ ❯❋❈●

❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛ ❉❡✉s✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ❢♦rç❛s ❡ ❝❛♣❛❝✐❞❛❞❡ ♣❛r❛ ❡①❡❝✉t❛r ❡ss❡ tr❛❜❛❧❤♦✳

❆♦s ♠❡✉s q✉❡r✐❞♦s ♣❛✐s✱ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞❡ ❢♦r♠❛ ♣rát✐❝❛ ♥❛ ❢♦r♠❛çã♦ ❞♦ ♠❡✉ ❝❛rát❡r✱ ❝♦♠ s❡✉s ❡①❡♠♣❧♦s ❞❡ ❞✐❣♥✐❞❛❞❡ ❡ ❤♦♥❡st✐❞❛❞❡✳

➚ ♠✐♥❤❛ ❛♠❛❞❛ ❡s♣♦s❛ ♣♦r t❡r ❛❜❞✐❝❛❞♦ ❞❡ ♠✐♥❤❛ ♣r❡s❡♥ç❛ ❡ ❛ss✉♠✐❞♦ ♠✐♥❤❛s r❡s♣♦♥s❛❜✐❧✐❞❛❞❡s ❢❛♠✐❧✐❛r❡s✳

❆♦s ♠❡✉s ✜❧❤♦s✱ ❝✉❥♦ ❛♠♦r ❡ ❝❛r✐♥❤♦ tr♦✉①❡r❛♠ ❛❧❡❣r✐❛ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❡①❛✉s✲ tã♦ ❡ ❞❡sâ♥✐♠♦✳

❆♦s ♠❡✉s ✐r♠ã♦s q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ♥❡ss❛ ❝❛♠✐♥❤❛❞❛✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧♦ ❝r❡s❝✐♠❡♥t♦ ✐♥t❡❧❡❝t✉❛❧ ♣r♦♣♦r❝✐✲ ♦♥❛❞♦ ❡ às ❝♦♦r❞❡♥❛çõ❡s✱ ❧♦❝❛❧ ❡ ❣❡r❛❧✱ ❞♦ P❘❖❋▼❆❚ ♣❡❧♦ ❡①❝❡❧❡♥t❡ ♣r♦❥❡t♦ q✉❡ ❝♦♥❝r❡t✐③♦✉ ❛ r❡❛❧✐③❛çã♦ ❞❡ ✉♠ s♦♥❤♦✳

❆ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r t♦❞❛ ❝♦♥tr✐❜✉✐çã♦ ✐♥t❡❧❡❝t✉❛❧ ❡ ♣❡❧❛s ❤♦r❛s ❞❡ ❡♠♣❡♥❤♦ ❡♠ ❜✉s❝❛ ❞❡ ♠❡ ♠♦str❛r ♦s ♠❡❧❤♦r❡s ❝❛♠✐♥❤♦s ❞❛ ♣❡sq✉✐s❛ ❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❝♦♠ ♠✐♥❤❛s ❧✐♠✐t❛çõ❡s ❡ ❞✐✜❝✉❧❞❛❞❡s q✉❡ ❢♦r❛♠ s✉♣❡r❛❞❛s ❝♦♠ s❡✉ ❡①❝❡❧❡♥t❡ ❛♣♦✐♦✳

❆♦s ❛♠✐❣♦s✱ ❆❧❞❡❝❦✱ ❆❧②ss♦♥✱ ❉✐❡❣♦✱ ❋r❛♥❝✐s❝♦ ❡ ▼❛r❝❡❧♦ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ❝✉rs♦ ❡ ❛s ❤♦r❛s ❞❡ ❡st✉❞♦s ❝♦♠♣❛rt✐❧❤❛❞❛s✳

➚ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✭❈❆P❊❙✮✱ ♣❡❧❛ ❜♦❧s❛ ❝♦♥❝❡❞✐❞❛✳

❉❡❞✐❝❛tór✐❛

❉❡s❞❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❤á ♠❛✐s ♦✉ ♠❡♥♦s ✹✵✵✵ ❛♥♦s✱ ✈ár✐♦s ♣♦✈♦s ❥á r❡s♦❧✈✐❛♠ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ♥♦ s❡✉ ❝♦t✐❞✐❛♥♦ ❛tr❛✈és ❞❡ ♣r♦❜❧❡♠❛s ❡ ❝♦♥str✉çõ❡s ♣rát✐❝❛s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛r❡♠♦s ❛❧❣✉♥s ♠ét♦❞♦s ❛❧❣é❜r✐❝♦s ❡ ❣❡♦♠étr✐❝♦s ✉s❛❞♦s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳ ■♥✐❝✐❛r❡♠♦s ❢❛❧❛♥❞♦ s♦❜r❡ ❢❛t♦r❛çã♦ ❡ ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐✱ r❡❧❛çõ❡s ❞❡ ●✐r❛r❞✱ t❡♦r❡♠❛ ❞❛s r❛í③❡s ❝♦♠♣❧❡①❛s ❡ ♦ t❡♦r❡♠❛ ❞❡ ♣❡sq✉✐s❛ ❞❛s r❛í③❡s r❛❝✐♦♥❛✐s✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ♠♦str❛r❡✲ ♠♦s ♦s ♠ét♦❞♦s ❛❧❣é❜r✐❝♦s ❞❡ ❱✐èt❡✱ ❈❛r❞❛♥♦✱ ❋❡rr❛r✐ ❡ ❊✉❧❡r✱ ❡ ❛❧❣✉♥s ♠ét♦❞♦s ❣❡♦♠étr✐❝♦s✱ ❝♦♠♦ ♦ ❞❛ ♣r♦♣♦rçã♦✱ ♦ ❞❡ ❉❡s❝❛rt❡s ❡ ❚❤♦♠❛s ❈❛r❧②❧❡ ❡ ❞❛s ❝ô♥✐❝❛s✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ✈❡r❡♠♦s ❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧✱ ♦ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ❞❡ ◆❡✇t♦♥✱ tr❛♥s❧❛çã♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✱ ♦ ✉s♦ ❞❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❢♦r♠❛ r❡❞✉③✐❞❛ ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❡ ❝♦♠ ❛✉①í❧✐♦ ❞❡ ❞❡r✐✈❛❞❛s ♠♦str❛r❡♠♦s ✉♠ ♠ét♦❞♦ ❞❡ r❡s♦❧✉çã♦ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞♦ ✸◦ _{❡ ✹}◦ _{❣r❛✉s✳}

P❛❧❛✈r❛s✲❝❤❛✈❡✿ P♦❧✐♥ô♠✐♦s✱ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳

❆❜str❛❝t

❙✐♥❝❡ ❛♥❝✐❡♥t t✐♠❡s✱ ❢♦r ❛❜♦✉t ✹✵✵✵ ②❡❛rs✱ ♠❛♥② ♣❡♦♣❧❡ ❤❛✈❡ ❛❧r❡❛❞② s♦❧✈❡❞ ♣♦❧②✲ ♥♦♠✐❛❧ ❡q✉❛t✐♦♥s ✐♥ t❤❡✐r ❞❛✐❧② ❧✐✈❡s t❤r♦✉❣❤ ♣r♦❜❧❡♠s ❛♥❞ ♣r❛❝t✐❝❡s ❝♦♥str✉❝t✐♦♥s✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ st✉❞② s♦♠❡ ❛❧❣❡❜r❛✐❝ ❛♥❞ ❣❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✉s❡❞ ❢♦r s♦❧✈✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s✳ ❲❡ st❛rt t❛❧❦✐♥❣ ❛❜♦✉t ❢❛❝t♦r✐♥❣ ❛♥❞ ❞✐✈✐s✐♦♥ ♦❢ ♣♦❧②♥♦♠✐✲ ❛❧s✱ ❞❡✈✐❝❡ ❇r✐♦t✲❘✉✣♥✐✱ r❡❧❛t✐♦♥s❤✐♣s ●✐r❛r❞✱ t❤❡♦r❡♠ ♦❢ t❤❡ ❝♦♠♣❧❡① r♦♦ts ❛♥❞ t❤❡ t❤❡♦r❡♠ ♦❢ t❤❡ r❛t✐♦♥❛❧ r♦♦ts r❡s❡❛r❝❤✳ ■♥ ❝❤❛♣t❡r ✷✱ ✇❡ ✇✐❧❧ s❤♦✇ t❤❡ ♠❡t❤♦❞s ❛❧❣❡❜r❛✐❝ ♦❢ ❱✐èt❡✱ ❈❛r❞❛♥♦✱ ❋❡rr❛r✐ ❛♥❞ ❊✉❧❡r✱ ❛♥❞ s♦♠❡ ❣❡♦♠❡tr✐❝ ♠❡t❤♦❞s✱ s✉❝❤ ❛s t❤❡ ♦❢ ♣r♦♣♦rt✐♦♥✱ ♦❢ t❤❡ ❉❡s❝❛rt❡s ❛♥❞ ❚❤♦♠❛s ❈❛r❧②❧❡ ❛♥❞ ♦❢ t❤❡ ❝♦♥✐❝❛s✳ ■♥ s❡❝t✐♦♥ ✸✱ ✇❡ s❡❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧✱ ◆❡✇t♦♥✬s ✐t❡r❛t✐✈❡ ♠❡t❤♦❞✱ tr❛♥s❧❛✲ t✐♦♥ ♦❢ ❝♦♦r❞✐♥❛t❡ ❛①❡s✱ ✉s✐♥❣ t❤❡ ❞❡r✐✈❡❞ ❢♦r t♦ ✜♥❞ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ r❡❞✉❝❡❞ ❢♦r♠ ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ✇✐t❤ t❤❡ ❛✐❞ ♦❢ ❞❡r✐✈❛t✐✈❡s s❤♦✇ ❛ ♠❡t❤♦❞ ♦❢ r❡s♦❧✉t✐♦♥ t❤❡ ❡q✉❛t✐♦♥s ✸r❞ ❛♥❞ ✹t❤ ❞❡❣r❡❡s✳

❑❡②✇♦r❞s✿ ♣♦❧②♥♦♠✐❛❧s✱ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s✳

■♥tr♦❞✉çã♦ ✈✐✐✐

✶ P♦❧✐♥ô♠✐♦s ❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✶

✶✳✶ P♦❧✐♥ô♠✐♦ ❡♠ ✉♠❛ ✈❛r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ❋✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ♣♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✸ Pr♦❞✉t♦ ♥♦tá✈❡✐s ❡ ❢❛t♦r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✹ ❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✺ ❉✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ♦✉ ❛❧❣é❜r✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✶ ❘❛✐③ ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ♦✉ ❛❧❣é❜r✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✷ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ á❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✸ ❘❡❧❛çõ❡s ❞❡ ●✐r❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✹ P❡sq✉✐s❛ ❞❡ r❛í③❡s r❛❝✐♦♥❛✐s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❞❡ ❝♦✲

❡✜❝✐❡♥t❡s ✐♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✺ ❘❛í③❡s ❝♦♠♣❧❡①❛s ♥ã♦ r❡❛✐s ♥✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❞❡ ❝♦❡✜✲

❝✐❡♥t❡s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷ ▼ét♦❞♦s ❞❡ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ✷✸

✷✳✶ ❊q✉❛çõ❡s ❞♦ ✶♦ _{❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸}

✷✳✶✳✶ ▼ét♦❞♦ ❛❧❣é❜r✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✶✳✷ ▼ét♦❞♦ ❣❡♦♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞♦ ✷♦ _{❣r❛✉ ❡ ❛ ❢ór♠✉❧❛ ❞❡ ❇❤ás❦❛r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻}

✷✳✷✳✶ ▼ét♦❞♦s ❛❧❣é❜r✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷✳✷ ▼ét♦❞♦s ❣❡♦♠étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ❊q✉❛çõ❡s ❞♦ ✸♦ _{❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾}

✷✳✸✳✶ ❙♦❧✉çã♦ ❞♦s ❜❛❜✐❧ô♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✸✳✷ ❆ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸✳✸ ❙♦❧✉çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❞❡ ❱✐èt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✳✹ ▼ét♦❞♦ ❣❡♦♠étr✐❝♦ ❞❛s ❝ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✹ ❊q✉❛çõ❡s ❞♦ ✹♦ _{❣r❛✉ ❡ ♦ ♠ét♦❞♦ ❞❡ ❋❡rr❛r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸}

✷✳✺ ▼ét♦❞♦ ❞❡ ❊✉❧❡r ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ ✹♦ _{❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻}

✷✳✻ ❖s ❝❛s♦s ✐♥út❡✐s ❞❛ ❡q✉❛çã♦ ❞♦ ✹♦ _{❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽}

✷✳✼ ❊q✉❛çõ❡s ❞♦ ✺♦ _{❣r❛✉✳ ❘✉✣♥✐✱ ❆❜❡❧ ❡ ●❛❧♦✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵}

✸ ❖ ✉s♦ ❞❡ ❞❡r✐✈❛❞❛s ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✻✶

✸✳✶ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✷ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ❡♥❝♦♥tr❛r r❛í③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✸ ❚r❛♥s❧❛çã♦ ❞❡ ❡✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✹ ❯s❛♥❞♦ ❞❡r✐✈❛❞❛s ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦s ♣♦❧✐♥ô♠✐♦s ♥❛

❢♦r♠❛ r❡❞✉③✐❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✸✳✹✳✶ ❆♣❧✐❝❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✸✳✺ ❯s❛♥❞♦ ❞❡r✐✈❛❞❛ ♣❛r❛ ❡s❝r❡✈❡r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐✲

❛✐s r❡❞✉③✐❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✻ ❘❡s♦❧✈❡♥❞♦ ❡q✉❛çõ❡s ❞♦ ✸♦ _{❡ ✹}◦ ❣r❛✉s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ❞❡r✐✈❛❞❛s ✳ ✳ ✳ ✼✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✾

❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞♦ ✉s♦ ❞❛s ❞❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛ ♦✉ ❝❛♥ô♥✐❝❛✱ ♦❜t✐❞♦s ❞❛ tr❛♥s❢♦r♠❛çã♦ x =y₋an−1/nan q✉❡ ❝♦♥✈❡rt❡ q✉❛❧q✉❡r ❡q✉❛çã♦ ❝♦♠♣❧❡t❛ ❞❡ ❣r❛✉

n ❞❛ ❢♦r♠❛ anxn+an−1xn−1+· · ·+a1x+a0 = 0 ❡♠ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❣r❛✉ n ❡♠

y ❢❛❧t❛♥❞♦ ♦ t❡r♠♦ ❞❡ ❡①♣♦❡♥t❡ n ₋1✱ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ ❢♦r♠❛ r❡❞✉③✐❞❛ ♦✉

❝❛♥ô♥✐❝❛✳

❆ ❜✉s❝❛ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❧❡✈♦✉ ♦s ❣ê♥✐♦s ❱✐èt✐✱ ❈❛r❞❛♥♦ ❡ ❋❡rr❛r✐ ❛ ❢❛③❡r ✉♠❛ tr❛♥s❧❛çã♦ ❞♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❣r❛✉ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❡❧❛ s❡❣✉✐♥t❡ s✉❜st✐t✉✐çã♦ x = y₋ b/na ♦♥❞❡ n é ♦ ❣r❛✉ ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧✳ ❖ ❣r❛♥❞❡ ♣r♦❜❧❡♠❛ ♥♦ ♥♦ss♦ ♣♦♥t♦ ❞❡ ✈✐st❛ sã♦

❛s r❡❧❛çõ❡s ❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❢♦r♠❛ r❡❞✉③✐❞❛ ❡ ❛ ❢♦r♠❛ ♦r✐❣✐♥❛❧ ✭❝♦♠♣❧❡t❛✮✳ ❖❜s❡r✈❡✿

◆❛ ❡q✉❛çã♦ ❝♦♠♣❧❡t❛ ❞♦ ✷♦ _{❣r❛✉ ax}2_{+bx+c= 0✱ ❝♦♠a 6= 0✱ ♠❡❞✐❛♥t❡ ❛ s✉❜s✲}

t✐t✉✐çã♦ ❞❡ x ♣♦r y₋b/2a ❛♣r❡s❡♥t❛ ❛ ❢♦r♠❛ r❡❞✉③✐❞❛ y2 _{+p = 0✱ ♦♥❞❡ ❛ r❡❧❛çã♦}

❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s é✿

p= 4ac−b

2

4a2 ✳

◆❛ ❡q✉❛çã♦ ❝♦♠♣❧❡t❛ ❞♦ ✸♦ _❣r❛✉ax3_+bx2_{+cx+d= 0✱ ❝♦♠ a6= 0✱ ♠❡❞✐❛♥t❡ ❛}

s✉❜st✐t✉✐çã♦ ❞❡ x ♣♦ry₋b/3a ❛♣r❡s❡♥t❛ ❛ ❢♦r♠❛ r❡❞✉③✐❞❛ y3_{+py+q = 0✱ ♦♥❞❡ ❛s}

r❡❧❛çõ❡s ❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s sã♦✿

p=₋ b

2

3a2 +

c

a✱

q= 2b

3

27a3 −

bc

3a2 +

d

a✳

◆❛ ❡q✉❛çã♦ ❝♦♠♣❧❡t❛ ❞♦ ✹♦_❣r❛✉ax4_+bx3_+cx2_{+dx+e = 0✱ ❝♦♠a6= 0✱ ♠❡❞✐❛♥t❡}

❛ s✉❜st✐t✉✐çã♦ ❞❡ x ♣♦ry₋b/4a ❛♣r❡s❡♥t❛ ❛ ❢♦r♠❛ r❡❞✉③✐❞❛ y4_+py2_{+qy+r = 0}

♦♥❞❡ ♣✱ q ❡ r sã♦✿

p= c

a −

3b2

8a2✱

q= d

a −

bc

2a2 +

b3

8a3✱

r= e

a −

bd

4a2 +

b2_c

16a3 −

3b4

256a4✳

❈♦♠♦ sã♦ ✐♥❞✐s♣❡♥sá✈❡✐s ♥♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ❝❛❞❛ ❡q✉❛çã♦✱ ✈❛♠♦s ♣r♦✲ ♣♦r ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡❧❛❝✐♦♥❛r ❞❡ ❢♦r♠❛ s✐♠♣❧❡s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛ ❝♦♠ ♦s ❞❛ ❢♦r♠❛ ♦r✐❣✐♥❛❧ ✭❝♦♠♣❧❡t❛✮✱ ✉s❛♥❞♦ ❛♣❡♥❛s ❛s ❞❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦✲ ♠✐❛❧ q✉❡ ♣♦ss✉✐ ❛s ♠❡s♠❛s r❛í③❡s q✉❡ ❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧ ✭❝♦♠♣❧❡t❛✮✳ ▼♦str❛r❡♠♦s t❛♠❜é♠ q✉❡ t♦❞❛ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛ ♦❜❡❞❡❝❡ ❛ ✉♠❛ ❢ór♠✉❧❛ ❣❡r❛❧✱ ❜❛s❡❛❞❛ ♥❛ sér✐❡ ❞❡ ❚❛②❧♦r✳

■♥✐❝✐❛r❡♠♦s ❢❛❧❛♥❞♦ s♦❜r❡ ♣♦❧✐♥ô♠✐♦s✱ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❡ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧✱ ❞❡♠♦♥str❛r❡♠♦s ♥❛ ♠❡❞✐❞❛ ❞♦ ♣♦ssí✈❡❧ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ t❡♦r❡✲ ♠❛s✳ ❊st✉❞❛r❡♠♦s ❛❧❣✉♥s ♠ét♦❞♦s ❛❧❣é❜r✐❝♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡st❛❝❛♥❞♦✱ ♦ ♠ét♦❞♦ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦s ❡ ❛ ❢ór♠✉❧❛ ❞❡ ❇❤ás❦❛r❛ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ ✷♦ _{❣r❛✉✱ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦ ❡ ❛ s♦❧✉çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❞❡ ❱✐èt❡ q✉❡}

❞✐s♣❡♥s❛ ♦ ✧❝❛❧❝✉❧♦ ❞❡ r❛í③❡s ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✧ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ ✸♦ _{❣r❛✉✱}

♦s ♠ét♦❞♦s ❞❡ ❋❡rr❛r✐ ❡ ❊✉❧❡r ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ ✹♦ _{❣r❛✉ ❡ ♦ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ❞❡}

◆❡✇t♦♥ ♣❛r❛ ❡q✉❛çõ❡s ❞❡ ❣r❛✉ ♠❛✐♦r q✉❡ ✹✳ ❱❡r❡♠♦s t❛♠❜é♠ ❛❧❣✉♥s ♠ét♦❞♦s ❣❡✲ ♦♠étr✐❝♦s q✉❡ ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞♦s ❢❛❝✐❧♠❡♥t❡ ♥❛s ❛✉❧❛s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❝♦♠♦ ♦s ♠ét♦❞♦s ❞❡ ❉❡s❝❛rt❡s ❡ ❚❤♦♠❛s ❈❛r❧②❧❡ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ ✷♦ _{❣r❛✉ ❡ ♦ ♠ét♦❞♦ ❞❛s}

❝ô♥✐❝❛s ♣❛r❛ ❛ ❞♦ ✸♦ _{❣r❛✉✳}

❊♠ ❝❛❞❛ s❡ssã♦ ❞♦s ❝❛♣ít✉❧♦s s❡rã♦ ❡①♣♦st❛s ❛ ✉t✐❧✐❞❛❞❡ ❞❡ ❝❛❞❛ ♠ét♦❞♦ ❡ té❝♥✐❝❛ ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✱ ♦❜❡❞❡❝❡♥❞♦ à s❡❣✉✐♥t❡ s❡q✉ê♥❝✐❛✿ ❢❛t♦r❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♣❡sq✉✐s❛ ❞❡ r❛í③❡s r❛❝✐♦♥❛s✱ ❡ ❡♠ ú❧t✐♠♦ ❝❛s♦✱ ❛s ❢ór♠✉❧❛s✳

P♦❧✐♥ô♠✐♦s ❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s

❊st✉❞❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ♦s ♠ét♦❞♦s ❡ té❝♥✐❝❛s ✉s❛❞❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳ ❉❡♠♦str❛r❡♠♦s ❛❧❣✉♠❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦✱ ❝♦♠♦ ❛ ❢❛t♦r❛çã♦ ❡ ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐✱ ❛s r❡❧❛çõ❡s ❞❡ ●✐r❛r❞✱ ♦ t❡♦r❡♠❛ ❞❛s r❛í③❡s ❝♦♠♣❧❡①❛s ❡ ♦ t❡♦r❡♠❛ ❞❡ ♣❡sq✉✐s❛ ❞❛s r❛í③❡s r❛❝✐♦♥❛✐s✳ P❛r❛ ✐ss♦ s❡❣✉✐r❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ❞♦s ❧✐✈r♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳

✶✳✶ P♦❧✐♥ô♠✐♦ ❡♠ ✉♠❛ ✈❛r✐á✈❡❧

❉❡✜♥✐çã♦ ✶✳✶ ❯♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ R ✭❝♦r♣♦ ❞♦s ♥ú♠❡r♦s

r❡❛✐s✮ ♥❛ ✈❛r✐á✈❡❧ x é ✉♠❛ ❡①♣r❡ssã♦ ❢♦r♠❛❧ ❞♦ t✐♣♦✿

p(x) =anxn+an−1xn−1+· · ·+a1x+a0,

♦♥❞❡ n _∈ N ❡ an, an−1, ..., a1, a0 sã♦ ♥ú♠❡r♦s r❡❛✐s✱ ❝❤❛♠❛❞♦s ❝♦❡✜❝✐❡♥t❡s ❞♦

♣♦❧✐♥ô♠✐♦✳ ❙❡ an 6= 0✱ ❞✐③❡♠♦s q✉❡ n é ♦ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦✳ ◆❡st❡ ❝❛s♦✱ an é

❝❤❛♠❛❞♦ ❞❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞♦ ♣♦❧✐♥ô♠✐♦✳

❖❜s❡r✈❛çã♦✿ ❯♠ ♣♦❧✐♥ô♠✐♦ s♦❜r❡ ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s C s❡ ❞❡✜♥❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳

❊①❡♠♣❧♦s✿

✶✳ p(x) = 5x₋4 é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞♦ ✶♦ ❣r❛✉ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s a1 = 5, a0 =−4.

✷✳ p(x) = 2x2_{−3x é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞♦ ✷}♦ _{❣r❛✉ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s a}

2 = 2✱ a1 =−3

❡ a0 = 0.

✸✳ p(x) = 3x3_{+ 2x}2_{+ 7x−3 é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞♦ ✸}♦ _{❣r❛✉ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s a} 3 = 3✱

a2 = 2✱ a1 = 7 ❡ a0 =−3.

⋄

✶✳✶✳ P❖▲■◆Ô▼■❖ ❊▼ ❯▼❆ ❱❆❘■➪❱❊▲

✶✳✶✳✶ ❋✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧

❉❡✜♥✐çã♦ ✶✳✷ ❯♠❛ ❢✉♥çã♦ ❢✿ R _→ R é ❞♦ t✐♣♦ ♣♦❧✐♥♦♠✐❛❧ q✉❛♥❞♦ ❡①✐st❡♠ ♥ú✲ ♠❡r♦s r❡❛✐s a0, a1, ..., an t❛✐s q✉❡✱ ♣❛r❛ t♦❞♦ x∈R t❡♠✲s❡

f(x) =anxn+an−1xn−1+· · ·+a1x+a0.

❖❜s❡r✈❛çõ❡s✿

✶✳ ❋✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s sã♦ ♦r✐❣✐♥❛❞❛s ♣♦r ♣♦❧✐♥ô♠✐♦s✳

✷✳ ❆ ❝❛❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛✲s❡ ✉♠ ú♥✐❝♦ ♣♦❧✐♥ô♠✐♦ ❡ ✈✐❝❡✲✈❡rs❛✱ ❞❡ ❢♦r♠❛ q✉❡ ♥ã♦ ❤á ❝♦♥❢✉sã♦ ❡♠ ♥♦s r❡❢❡r✐r♠♦s s❡♠ ❞✐st✐♥çã♦ às ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ♦✉ ❛♦s ♣♦❧✐♥ô♠✐♦s✳

❊①❡♠♣❧♦s✿

✶✳ ❆s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞♦ t✐♣♦p(x) =a✱ ❝♦♠a₆= 0✱ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❢✉♥çõ❡s

❝♦♥st❛♥t❡s✳

✷✳ ❆s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❛ ❢♦r♠❛ p(x) = ax+b✱ ❝♦♠ a₆= 0✱ sã♦ ❝❤❛♠❛❞❛s ❞❡

❢✉♥çõ❡s ❛✜♥s✳

✸✳ ❆s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞♦ t✐♣♦p(x) = ax2_{+bx+c✱ ❝♦♠a6= 0✱ sã♦ ❝❤❛♠❛❞❛s}

❞❡ ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✳

⋄

P♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦

❉❡✜♥✐çã♦ ✶✳✸ ❯♠ ♣♦❧✐♥ô♠✐♦ ❝✉❥♦s ❝♦❡✜❝✐❡♥t❡s sã♦ t♦❞♦s ✐❣✉❛✐s ❛ ③❡r♦ é ❞❡♥♦♠✐✲ ♥❛❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ♦✉ ♣♦❧✐♥ô♠✐♦ ③❡r♦✳ ❉❡ ♠♦❞♦ q✉❡✱ ✉♠ ♣♦❧✐♥ô♠✐♦ p(x) = anxn +an−1xn−1 +· · · +a1x+a0 é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ s❡

an=an−1 =· · ·=a1 =a0 = 0✳

❖❜s❡r✈❛çã♦✿ ❖ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ♥ã♦ é ❞❡✜♥✐❞♦✳

■❣✉❛❧❞❛❞❡ ❞❡ ♣♦❧✐♥ô♠✐♦s

❉❡✜♥✐çã♦ ✶✳✹ ❉❛❞♦s ❞♦✐s ♣♦❧✐♥ô♠✐♦s p1(x) = anxn+an−1xn−1 +· · ·+a1x+a0 ❡

p2(x) =bmxm+bn−1xn−1 +· · ·+b1x+a0✳ ❊♥tã♦ p1(x) =p2(x) s❡✱ m =n ❡✱ ❛❧é♠

❞✐ss♦✱ ♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s sã♦ ♦r❞❡♥❛❞❛♠❡♥t❡ ✐❣✉❛✐s✱ ♦✉ s❡❥❛✿

an =bm, an−1 =bm−1, ..., a1 =b1, a0 =b0.

❊①❡♠♣❧♦✿ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ❞❡ a✱ b✱ c✱ d ❡ e ❞❡ ♠♦❞♦ q✉❡ ♦s ♣♦❧✐♥ô♠✐♦s p(x) =ax4_{+ 5x}2_{+dx−b ❡q(x) = 2x}4_{+ (b−3)x}3_{+ (2c−1)x}2_{+x+es❡❥❛♠ ✐❣✉❛✐s✳}

❙♦❧✉çã♦✿ P❛r❛ q✉❡ s❡❥❛ p(x) = q(x)✱ ❞❡✈❡♠♦s t❡r✿

a= 2✱

0 =b₋3_⇒b = 3✱

5 = 2c₋1_⇒2c= 6 _⇒c= 3✱

d= 1✱

e=₋b _⇒e=₋3✳

▲♦❣♦✱ a = 2, b= 3, c= 3, d= 1 ❡e =₋3✳ _⋄

❱❛❧♦r ♥✉♠ér✐❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧

❉❡✜♥✐çã♦ ✶✳✺ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧f(x)❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ α✳❖ ✈❛❧♦r

♥✉♠ér✐❝♦ ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧f(x)♣❛r❛x=αé ♦ ✈❛❧♦r q✉❡ s❡ ♦❜té♠ s✉❜st✐t✉✐♥❞♦ x ♣♦r α ❡ ❡❢❡t✉❛♥❞♦ ♦s ❝á❧❝✉❧♦s ♥❡❝❡ssár✐♦s✳ ■♥❞✐❝❛✲s❡ ♣♦r f(α)✳ ❊♥tã♦✱ f(α) é ♦

✈❛❧♦r ♥✉♠ér✐❝♦ ❞❡ f(x) ♣❛r❛ x=α✳ ❆ss✐♠✱ ❞❡ ♠♦❞♦ ❣❡r❛❧✱ ❞❛❞♦ ♦ ♣♦❧✐♥ô♠✐♦

f(x) = anxn+an−1xn−1+· · ·+a1x+a0

♦ ✈❛❧♦r ❞❡ f(x) ♣❛r❛ x=α é

f(α) =anαn+an−1αn−1+· · ·+a1α+a0.

❊①❡♠♣❧♦✿ ❖ ✈❛❧♦r ♥✉♠ér✐❝♦ ❞❡ p(x) = 2x2_{−3x+ 5 ♣❛r❛ x= 4 é}

p(4) = 2(4)2₋3(4) + 5 = 32₋12 + 5 = 25.

⋄

❖❜s❡r✈❛çã♦✿ ❙❡ f(α) = 0✱ ♦ ♥ú♠❡r♦ α é ❞❡♥♦♠✐♥❛❞♦ r❛✐③ ❞❡ f(x)✳ P♦r ❡①❡♠♣❧♦✱

♥❛ ❢✉♥çã♦f(x) = x2_{−6x+8✱ t❡♠♦sf(2) = 0✳ ▲♦❣♦✱ ✷ é r❛✐③ ❞❡ss❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧✳}

●rá✜❝♦s ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s

❖s ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s sã♦ ❡st✉❞❛❞♦s ❞❡s❞❡ ♦ ✾♦ _{❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥✲}

❞❛♠❡♥t❛❧✱ ❛♣r❡s❡♥t❛♥❞♦ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ♥❛ ❋ís✐❝❛✱ ♥❛ ◗✉í♠✐❝❛ ❡ ♥❛ ❊st❛tíst✐❝❛✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ❈✐♥❡♠át✐❝❛ ♦ ♠♦✈✐♠❡♥t♦ r❡t✐❧í♥❡♦ ✉♥✐❢♦r♠❡ ✭▼❘❯✮ t❡♠ ❛ ♣♦s✐çã♦

✶✳✶✳ P❖▲■◆Ô▼■❖ ❊▼ ❯▼❆ ❱❆❘■➪❱❊▲

❞❡ ✉♠ ♠ó✈❡❧ ❞❡s❝r✐t❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ❛✜♠ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ♣♦r ✉♠❛ ❢✉♥çã♦ ❝♦♥s✲ t❛♥t❡✳ ❏á ♥♦ ♠♦✈✐♠❡♥t♦ r❡t✐❧í♥❡♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ✈❛r✐❛❞♦ ✭▼❘❯❱✮✱ ❛ ♣♦s✐çã♦ ❞❡ ✉♠ ♠ó✈❡❧ é ❞❡s❝r✐t❛ ♣♦r ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ♣♦r ✉♠❛ ❢✉♥çã♦ ❛✜♠ ❡ ❛ ❛❝❡❧❡r❛çã♦ ♣♦r ✉♠ ❢✉♥çã♦ ❝♦♥st❛♥t❡✳ ❱❡r❡♠♦s ❛❣♦r❛ ✉♠❛s ❞✐❝❛s s✐♠♣❧❡s ❞❡ ❝♦♠♦ ❝♦♥str✉✐r ❡ss❡s ❣rá✜❝♦s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳

❋✉♥çã♦ ❝♦♥st❛♥t❡✳

❙❡❥❛ f(x) : R _→ R ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡✜♥✐❞❛ ♣♦r f(x) = a0✱ ❝♦♠ a0 6= 0✱

❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢✉♥çã♦ ❝♦♥st❛♥t❡✱ ♦ s❡✉ ❣rá✜❝♦ é ✉♠❛ ❧✐♥❤❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧✳ ❉❛ ❣❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ s❛❜❡♠♦s q✉❡ ♣♦r ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ♣❛ss❛ ✉♠❛ ú♥✐❝❛ ❧✐♥❤❛ r❡t❛❀ ❜❛st❛ ❝❛❧❝✉❧❛r ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❞❛ r❡t❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s

(x, f(x)) ❡ tr❛ç❛r ❛ ❧✐♥❤❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❡ss❡s ♣♦♥t♦s✳

❊①❡♠♣❧♦✿ ❊s❜♦❝❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f(x) = 4✳

❙♦❧✉çã♦✿ ❊s❝♦❧❤❡♥❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ x = 0 _⇒ f(0) = 4 ❡ x = 4 _⇒ f(4) = 4✳

▲♦❣♦✱ t❡♠♦s ♦s ♣♦♥t♦s P1 = (0,4) ❡ P2 = (4,4) ♣♦r ♦♥❞❡ ✈❛♠♦s tr❛ç❛r ♥♦ss❛ ❧✐♥❤❛

r❡t❛✳ ❱❡r ✜❣✉r❛ ✶✳✶✳ ⋄

❋✐❣✉r❛ ✶✳✶✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f(x) = 4✳

❋✉♥çã♦ ❛✜♠✳

❙❡❥❛f(x) :R_→R✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡✜♥✐❞❛ ♣♦r f(x) = ax+b ❝♦♠a ₆= 0

❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢✉♥çã♦ ❛✜♠✳ ❊❧❛ s❡rá ❝r❡s❝❡♥t❡ s❡ a >0 ❡ ❞❡❝r❡s❝❡♥t❡ s❡ a <0✱ ♦

s❡✉ ❣rá✜❝♦ é ✉♠❛ ❧✐♥❤❛ r❡t❛ ✐♥❝❧✐♥❛❞❛ ❛♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ à ❢✉♥çã♦ ❝♦♥st❛♥t❡✱ ❜❛st❛ ❝❛❧❝✉❧❛r ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❞❛ r❡t❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s (x, f(x)) ❡

tr❛ç❛r ❛ ❧✐♥❤❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❡ss❡s ♣♦♥t♦s✳

❊①❡♠♣❧♦✿ ❊s❜♦❝❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❛✜♠ f(x) = 2x₋4✳

❙♦❧✉çã♦✿ ❈♦♠♦ a = 2 > 0✱ ❛ ❢✉♥çã♦ é ❝r❡s❝❡♥t❡✳ ❊s❝♦❧❤❡♥❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡

x = 0 ❡ x= 4 ♦❜t❡♠♦s f(0) =₋4 ❡ f(4) = 4✳ ▲♦❣♦✱ t❡♠♦s ♦s ♣♦♥t♦s P1 = (0,−4)

❡ P2 = (4,4)✱ ♣♦r ♦♥❞❡ ✈❛♠♦s tr❛ç❛r ♥♦ss❛ ❧✐♥❤❛ r❡t❛✳ ❱❡r ✜❣✉r❛ ✶✳✷✳ ⋄

❋✐❣✉r❛ ✶✳✷✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❛✜♠f(x) = 2x₋4✳

❋✉♥çã♦ q✉❛❞rát✐❝❛✳

❙❡❥❛ f(x) : R _→ R ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡✜♥✐❞❛ ♣♦r f(x) = ax2 _+bx+c✱

❝♦♠ a ₆= 0✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ ❖ s❡✉ ❣rá✜❝♦ é ✉♠❛ ♣❛rá❜♦❧❛ ❝♦♠

❝♦♥❝❛✈✐❞❛❞❡ ♣❛r❛ ❝✐♠❛ s❡ a > 0 ❡✱ ♣❛r❛ ❜❛✐①♦ s❡ a < 0✳ P♦ss✉✐ ✉♠ ♣♦♥t♦ ❡s♣❡❝✐❛❧

❝❤❛♠❛❞♦ ❞❡ ✈ért✐❝❡ ❞❡ ❝♦♦r❞❡♥❛❞❛sV =

−₂b_a,₋b

2_−4ac

4a

✳ ▼❛r❝❛♥❞♦ ♦ ✈ért✐❝❡✱ ❡ ❛❧❣✉♥s ♣♦♥t✐♥❤♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s (x, f(x))♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ♣♦❞❡♠♦s tr❛ç❛r ♣♦r

✶✳✶✳ P❖▲■◆Ô▼■❖ ❊▼ ❯▼❆ ❱❆❘■➪❱❊▲

❡❧❡s ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳

❊①❡♠♣❧♦✿ ❊s❜♦❝❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ f(x) =x2_{−6x+ 5✳}

❙♦❧✉çã♦✿ ❆s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ sã♦ V =

−₂b_a,₋b

2_−4ac

4a

= (3,₋4)✳ ❈♦♠♦

a = 1 > 0 ❛ ❝♦♥❝❛✈✐❞❛❞❡ ❞❛ ♣❛rá❜♦❧❛ é ♣❛r❛ ❝✐♠❛✳ ❈❛❧❝✉❧❡♠♦s ❛❧❣✉♥s ♣♦♥t♦s ❞❡

❝♦♦r❞❡♥❛❞❛s (x, f(x))✳ P❛r❛ x = 1 _⇒ f(1) = 0❀ ♣❛r❛ x = 2 _⇒ f(2) = 3❀ ♣❛r❛

x = 4_⇒ f(4) = 3 ❡ ♣❛r❛ x = 5_⇒ f(5) = 0✳ ▼❛r❝❛♥❞♦ ♦ ✈ért✐❝❡ ❡ ❡ss❡s ♣♦♥t♦s ♥♦

♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ♣♦❞❡♠♦s tr❛ç❛r✱ ♣♦r ❡❧❡s✱ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ❱❡r ❋✐❣✉r❛ ✶✳✸✳ ⋄

❋✐❣✉r❛ ✶✳✸✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ f(x) =x2_{−6x+ 5✳}

✶✳✶✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ♣♦❧✐♥ô♠✐♦s

❆❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s

❉❡✜♥✐çã♦ ✶✳✻ ❆ ❛❞✐çã♦ ❞❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s é ❢❡✐t❛ s♦♠❛♥❞♦ ♦s t❡r♠♦s ❞❡ ♠❡s♠♦ ❡①♣♦❡♥t❡✳

❉❛❞♦s p(x) = n X

i=0

aixi ❡ g(x) =

n X

i=0

bixi✱ t❡♠♦s q✉❡✿

p(x) +g(x) =

n X

i=0

aixi !

+ n X

i=0

bixi !

= n X

i=0

(ai+bi)xi.

❊①❡♠♣❧♦s✿

✶✳ ❉❛❞♦sp(x) = 3x2_{+ 2x+ 1❡q(x) = −x}3_{+ 4x}2_{−2x−5✱ ❞❡t❡r♠✐♥❡p(x) +q(x)✳}

❙♦❧✉çã♦✿ ❈♦♠♣❧❡t❛♥❞♦ ♦ ♣♦❧✐♥ô♠✐♦ p(x) ❝♦♠ ♦ t❡r♠♦ 0x3_{, t❡♠♦s✿}

p(x) +q(x) = (0x3_{+ 3x}2_{+ 2x+ 1) + (−x}3_{+ 4x}2_−2x−5)

= (0₋1)x3_{+ (3 + 4)x}2_{+ (2−2)x+ (1−5)}

=₋x3_{+ 7x}2_−4✳

✷✳ ❉❛❞♦sp(x) = 6x3_+3x2_{−2x+4❡q(x) =−3x}3_+3x3_{−x+1✱ ❝❛❧❝✉❧❡p(x)+q(x)✳}

❙♦❧✉çã♦✿ ❈♦♠♦ p(x) ❡q(x)♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❣r❛✉✱ t❡♠♦s✿

p(x) +q(x) = (6x3_{+ 3x}2_{−2x+ 4) + (−3x}3_{+ 3x}3_{−x+ 1)}

= (6₋3)x3_{+ (3 + 3)x}2_{+ (−2−1)x+ (4 + 1)}

= 3x3_{+ 6x}2_{−3x+ 5✳}

⋄

▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s

❉❡✜♥✐çã♦ ✶✳✼ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s é ❢❡✐t❛ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❝❛❞❛ t❡r♠♦ ❞♦ ♣r✐♠❡✐r♦ ♣♦❧✐♥ô♠✐♦ ♣♦r t♦❞♦s ♦s t❡r♠♦s ❞♦ s❡❣✉♥❞♦ ❡ s♦♠❛♥❞♦ ❡ss❡s ♣r♦❞✉t♦s✳

❉❛❞♦s p(x) = m X

i=0

aixi ❡ q(x) =

n X

j=0

bjxj✱ t❡♠♦s✿

p(x)_·q(x) =

m X

i=0

aixi !

· n X

j=0

bjxj !

= m+n X

k=0

X

i+j=k

ai·bj !

xk.

✶✳✶✳ P❖▲■◆Ô▼■❖ ❊▼ ❯▼❆ ❱❆❘■➪❱❊▲

❊①❡♠♣❧♦s✿

✶✳ ❙❡♥❞♦ p(x) = 3x2 _{❡q(x) =x}2_{−3x+ 2✱ ❝❛❧❝✉❧❡ p(x)·q(x)✳}

❙♦❧✉çã♦✿ ▼✉❧t✐♣❧✐❝❛♥❞♦ t❡r♠♦ ❛ t❡r♠♦✱ t❡♠♦s✿

p(x)_·q(x) = (3x2_)(x2_{−3x+ 2) = 2·3x}2_−3x·3x2_+x2_·3x2 _{= 6x}2_−9x3_{+ 3x}4_✳

✷✳ ❉❛❞♦s ♦s ♣♦❧✐♥ô♠✐♦s p(x) = 3x₋4 ❡ q(x) =₋2x+ 5✱ ❝❛❧❝✉❧❡ p(x)_·q(x)✳

❙♦❧✉çã♦✿ ▼✉❧t✐♣❧✐❝❛♥❞♦ t❡r♠♦ ❛ t❡r♠♦✱ t❡♠♦s✿

p(x)_·q(x) = (3x₋4)(₋2x+ 5) =₋20 + 15x+ 8x₋6x2 _{=−20 + 23x−6x}2_✳

⋄

✶✳✶✳✸ Pr♦❞✉t♦ ♥♦tá✈❡✐s ❡ ❢❛t♦r❛çã♦

◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ tr❛❜❛❧❤❛r❡♠♦s ❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ♦♥❞❡ ❞✐s❝✉t✐r❡♠♦s ❛ s✉❛ r❡s♦❧✉çã♦ ♣♦r ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ q✉❛❞r❛❞♦s ♦✉ ♣♦r ❢❛t♦r❛çã♦✱ q✉❛♥❞♦ ❢♦r ♦ ❝❛s♦✳ ❚❛♠❜é♠ ✉s❛r❡♠♦s ♣r♦❞✉t♦s ♥♦tá✈❡✐s ❡♠ ♦✉tr❛s ❞❡♠♦♥str❛çõ❡s✳

Pr♦❞✉t♦s ♥♦tá✈❡✐s

❙ã♦ ♣r♦❞✉t♦s q✉❡ ❛♣❛r❡❝❡♠ ❝♦♠ ♠✉✐t❛ ❢r❡q✉ê♥❝✐❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❡①♣r❡ssõ❡s ❛❧❣é❜r✐❝❛s✿

• ◗✉❛❞r❛❞♦ ❞❛ s♦♠❛ ❞❡ ❞♦✐s t❡r♠♦s✿

(a+b)2 = (a+b)(a+b) = a2+ab+ba+b2 =a2+2ab+b2 _⇒(a+b)2 =a2+2ab+b2.

• ◗✉❛❞r❛❞♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s t❡r♠♦s✿

(a₋b)2 = (a₋b)(a₋b) = a2₋ab₋ba+b2 =a2₋2ab+b2 _⇒(a₋b)2 =a2₋2ab+b2.

• Pr♦❞✉t♦ ❞❛ s♦♠❛ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s t❡r♠♦s✿

(a+b)(a₋b) =a2₋ab+ba+b2 =a2₋b2 _⇒(a+b)(a₋b) =a2₋b2.

• ❈✉❜♦ ❞❛ s♦♠❛ ❞❡ ❞♦✐s t❡r♠♦s ✿

(a+b)3 =a3+ 3a2b+ 3ab2+b3.

• ❈✉❜♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s t❡r♠♦s✿

(a₋b)3 =a3₋3a2b+ 3ab2₋b3.

• ❙♦♠❛ ❞❡ ❞♦✐s ❝✉❜♦s✿

(a+b)(a2₋ab+b2) = a3+b3.

• ❉✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s ❝✉❜♦s

(a₋b)(a2 +ab+b2) =a3₋b3.

❖❜s❡r✈❛çã♦✿ ❈❛s♦ s❡❥❛ ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ♦ ♣r♦❞✉t♦ ❞❡ ✉♠ ❜✐♥ô♠✐♦ ❞❡ ♣♦tê♥❝✐❛ ♠❛✐♦r q✉❡ ✸✱ ♣♦❞❡♠♦s ✉s❛r ♦ ❢❛♠♦s♦ ❜✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥

(a+b)n = n X

k=0

(nk)a n−k

bk

❝♦♠ k✱ n_∈N✳

❋❛t♦r❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✿

❆ ❢❛t♦r❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s é ❞❡ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦✲ ♠✐❛✐s✱ ❢❛❝✐❧✐t❛♥❞♦ ❛ s✐♠♣❧✐✜❝❛çã♦ ❞❡ ❡①♣r❡ssõ❡s ❛❧❣é❜r✐❝❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♠✐t❡s✳

❱❡r❡♠♦s ❛❣♦r❛ ❛❧❣✉♥s ❝❛s♦s✿

• ❋❛t♦r❛çã♦ ♣♦r ❢❛t♦r ❝♦♠✉♠✿ ❉❡✈❡✲s❡ ♦❜s❡r✈❛r s❡ t♦❞♦s ♦ t❡r♠♦s ❞♦ ♣♦❧✐♥ô✲

♠✐♦ ❛♣r❡s❡♥t❛♠ ✉♠ ❢❛t♦r ❡♠ ❝♦♠✉♠✳ ❊♠ ❝❛s♦ ❛✜r♠❛t✐✈♦ ❞❡✈❡♠♦s ❝♦❧♦❝á✲❧♦ ❡♠ ❡✈✐❞ê♥❝✐❛✳ P♦r ❡①❡♠♣❧♦✳

4x3₋8x2+ 16x= 4x(x2₋2x+ 4).

• ❋❛t♦r❛çã♦ ♣♦r ❛❣r✉♣❛♠❡♥t♦✿ ❉❡✈❡✲s❡ ❛♣❧✐❝❛r ❛ ❢❛t♦r❛çã♦ ♣♦r ❢❛t♦r ❝♦♠✉♠

♠❛✐s ❞❡ ✉♠❛ ✈❡③✳ ❱❡❥❛ ♦ ❡①❡♠♣❧♦✿

2x3+ 4x2₋6x₋12 = 2x2(x+ 2)₋6(x+ 2) = (x+ 2)(2x2₋6).

• ❋❛t♦r❛çã♦ ❞❡ ✉♠ tr✐♥ô♠✐♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✿

a2+ 2ab+b2 = (a+b)2

✶✳✶✳ P❖▲■◆Ô▼■❖ ❊▼ ❯▼❆ ❱❆❘■➪❱❊▲

• ❋❛t♦r❛çã♦ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

a2₋b2 = (a+b)(a₋b)

• ❋❛t♦r❛çã♦ ♣❡❧❛ s♦♠❛ ❞❡ ❞♦✐s ❝✉❜♦s✳

a3+b3 = (a+b)(a2₋ab+b2)

• ❋❛t♦r❛çã♦ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s ❝✉❜♦s✳

a3 ₋b3 = (a+b)(a2+ab+b2).

❈♦♠♦ ❛♣❧✐❝❛çã♦✱ ✈❡r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞♦ ✉s♦ ❞❡ ❢❛t♦r❛çã♦ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳

✶✳ ❘❡s♦❧✈❛ ❛ ❡q✉❛çã♦ ❞♦ ✸♦ _{❣r❛✉ 9x}3 _−27x2 _{−4x+ 12 = 0 ♣♦r ♠❡✐♦ ❞❡ ❢❛t♦r❛✲}

çã♦✳

❙♦❧✉çã♦✿ ❋❛③❡♥❞♦ ❛ ❢❛t♦r❛çã♦ ♣♦r ❛❣r✉♣❛♠❡♥t♦✱ t❡♠♦s

9x3₋27x2 ₋4x+ 12 = 9x2(x₋3)₋4(x₋3) = (x₋3)(9x2₋4).

❋❛t♦r❛♥❞♦ ♦ s❡❣✉♥❞♦ ❢❛t♦r✱ q✉❡ é ✉♠❛ ❞✐❢❡r❡♥ç❛ ❞❡ q✉❛❞r❛❞♦s✱ t❡♠♦s

9x3₋27x2₋4x+ 12 = (x₋3)(9x2₋4) = (x₋3)(3x+ 2)(3x₋2)

q✉❡ s✉❜st✐t✉í❞❛ ♥❛ ❡q✉❛çã♦ ✜❝❛✿

9x3₋27x2₋4x+ 12 = (x₋3)(3x+ 2)(3x₋2) = 0

q✉❡ ✐♠♣❧✐❝❛

x₋3 = 0 _⇒x= 3

♦✉

3x+ 2 = 0_⇒x=₋2 3

♦✉

3x₋2 = 0_⇒x= 2 3.

P♦rt❛♥t♦✱ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ 9x3_−27x2 _{−4x+ 12 = 0 sã♦✿ ✸✱} 2

3 ❡− 2 3✳

✷✳ ❊♥❝♦♥tr❡ t♦❞❛s ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ 4x4_{+ 32x= 0✳}

❙♦❧✉çã♦✿ ❋❛③❡♥❞♦ ❛ ❢❛t♦r❛çã♦ ♣♦r ❢❛t♦r ❝♦♠✉♠✱ t❡♠♦s

4x4+ 32x= 4x(x3+ 8).

❋❛t♦r❛♥❞♦ ♦ s❡❣✉♥❞♦ ❢❛t♦r✱ q✉❡ é ❛ s♦♠❛ ❞❡ ❞♦✐s ❝✉❜♦s✱ t❡♠♦s✿

4x4_{+ 32x= 4x(x}3_{+ 8) = 4x(x+ 2)(x}2_{−2x+ 4),}

q✉❡ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦✱ ✜❝❛✿

4x4+ 32x= 4x(x3+ 8) = 4x(x+ 2)(x2₋2x+ 4) = 0,

q✉❡ ✐♠♣❧✐❝❛

4x= 0 _⇒x= 0

♦✉

x+ 2 = 0_⇒x=₋2

♦✉

x2₋2x+ 4 = 0_⇒  



x1 = 1 +

√ 3i

x2 = 1−

√ 3i.

P♦rt❛♥t♦✱ ❛s q✉❛tr♦ r❛í③❡s ❞❛ ❡q✉❛çã♦ sã♦4x4_{+32x= 0sã♦✿ 0,−2✱1+}√_3i❡1−√_3i✳

❙❡r✐❛ ♠✉✐t♦ ❜♦♠ q✉❡ t♦❞❛s ❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ♣✉❞❡ss❡♠ s❡r r❡s♦❧✈✐❞❛s ♣♦r ❢❛t♦r❛çã♦✱ ♣♦ré♠✱ ♥❡♠ s❡♠♣r❡ é ❢á❝✐❧ ❡♥①❡r❣❛r ✉♠ ❢❛t♦r ❝♦♠✉♠ ♣❛r❛ s❡ ❡❢❡t✉❛r ❡♠ s❡❣✉✐❞❛ ♦ ❛❣r✉♣❛♠❡♥t♦ ♦✉ ♦✉tr♦ t✐♣♦ ❞❡ ❢❛t♦r❛çã♦✳ ❊st✉❞❛r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡ssã♦ ♦ t❡♦r❡♠❛ ❞❛s r❛í③❡s r❛❝✐♦♥❛✐s q✉❡ ♥♦s ❛❥✉❞❛rá ❛ ❡♥❝♦♥tr❛r r❛í③❡s r❛❝✐♦♥❛✐s✱ ❝❛s♦ ❡①✐st❛♠✳

✶✳✶✳✹ ❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s

❚❡♦r❡♠❛ ✶✳✽ ✭❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✮✳ ❙❡❥❛♠ p(x) ❡d(x) ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❝♦♠

❝♦❡✜❝✐❡♥t❡s r❡❛✐s✱ ❝♦♠ d(x)₆= 0✳ ❊♥tã♦✱ ❡①✐st❡♠ ú♥✐❝♦s ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s

r❡❛✐s q(x) ❡ r(x) t❛✐s q✉❡✿

p(x) =d(x)q(x)+r(x),♦♥❞❡r(x) = 0♦✉ ♦ ❣r❛✉ ❞❡r(x)é ♠❡♥♦r q✉❡ ♦ ❣r❛✉ ❞❡d(x).

❖s ♣♦❧✐♥ô♠✐♦s q(x) ❡ r(x) sã♦ ❝❤❛♠❛❞♦s ❞❡ q✉♦❝✐❡♥t❡ ❡ r❡st♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

✶✳✶✳ P❖▲■◆Ô▼■❖ ❊▼ ❯▼❆ ❱❆❘■➪❱❊▲

❉❡♠♦♥str❛çã♦✿

Pr♦✈❡♠♦s ♣r✐♠❡✐r♦ ❛ ❡①✐stê♥❝✐❛✳ ❙❡❥❛♠ p(x) = anxn+an−1xn−1+· · ·+a1x+a0✱

❝♦♠ an 6= 0 ❡ d(x) = bpxp+bp−1xp−1+· · ·+b1x+b0✳ ❙❡ n < p t♦♠❡✱ q(x) = 0 ❡

r(x) =p(x)✳ ❆❣♦r❛✱ s❡ n _≥ p ❡ bp 6= 0✱ t♦♠❡ q0(x) = a_bn

px

n−p_{✳ ❱❡♠♦s q✉❡ ♦ ♣♦❧✐♥ô✲}

♠✐♦p(x)₋d(x)q0(x) = (an−1−a_bn

pbp−1)x

n−1_{+· · · t❡♠✱ ♥♦ ♠á①✐♠♦✱ ❣r❛✉ ✐❣✉❛❧ ❛n−1✳}

❋❛③❡♥❞♦ ♦ ♠❡s♠♦ ♣r♦❝❡ss♦ ♣❛r❛ p(x) ₋ d(x)q0(x)✱ ✈❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô✲

♠✐♦ q1(x) t❛❧ q✉❡p(x)−d(x)q0(x)−d(x)q1(x) =p(x)−d(x)[q0(x) +q1(x)] t❡♠✱ ♥♦

♠á①✐♠♦✱ ❣r❛✉ n₋2✳

Pr♦ss❡❣✉✐♥❞♦✱ ✈❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ q(x) = [q0(x) +q1(x)] +· · ·+qn−p(x)

t❛❧ q✉❡ p(x)₋d(x)q(x) t❡♠ ❣r❛✉ ♥♦ ♠á①✐♠♦ p₋1✳ ❈❤❛♠❛♥❞♦ p(x)₋d(x)q(x) ❞❡

r(x)✱ ❡stá ♣r♦✈❛❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ q(x) ❡ r(x)✳

❯♥✐❝✐❞❛❞❡✱ s❡ p(x) = d(x)q1(x) +r1(x) ❡ p(x) = d(x)q2(x) + r2(x) ❝♦♠ ♦s

❣r❛✉s ❞❡ r1(x) ❡ r2(x) ❛♠❜♦s ♠❡♥♦r❡s q✉❡ ♦ ❣r❛✉ ❞❡ d(x)✱ t❡♠♦s✱ s✉❜tr❛✐♥❞♦✱ q✉❡

d(x)[q1(x)−q2(x)] = r1(x)−r2(x)✳ ❙❡ q1(x)−q2(x) ♥ã♦ ❢♦r ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✱ ♦

❣r❛✉ ❞♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ s❡rá ♠❛✐♦r ♦✉ ✐❣✉❛❧ q✉❡ ♦ ❣r❛✉ ❞❡ d(x) ✳ P♦r ♦✉tr♦ ❧❛❞♦✱

♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ t❡♠ ❣r❛✉ ♠❡♥♦r q✉❡ d(x)✳ ▲♦❣♦✱ q1(x)−q2(x) é ✐❞❡♥t✐❝❛♠❡♥t❡

♥✉❧♦✱ ♦✉ s❡❥❛✱ q1(x) = q2(x)✱ q✉❡ ✐♠♣❧✐❝❛ r1(x) =r2(x)✳

❊①❡♠♣❧♦✿ ❱❛♠♦s ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞♦ ♣♦❧✐♥ô♠✐♦p(x) =x3+ 10♣♦rd(x) = x+ 2✳

❙♦❧✉çã♦✿ ❈♦♠♦ ♦ ❣r❛✉ ❞❡ p(x) é ✸ ❡ ♦ ❣r❛✉ ❞❡ d(x) é ✶✱ ♦ ❣r❛✉ ❞❡ q(x) é ♥♦

♠á①✐♠♦ ✷✱ ♦✉ s❡❥❛✱ q(x) =ax2 +bx+c ❡r(x) =r é ✉♠❛ ❝♦♥st❛♥t❡✳

❉❛ ✐❞❡♥t✐❞❛❞❡ p(x) = d(x)q(x) +r(x)✱ t❡♠✲s❡

x3_{+10 = (x+2)(ax}2_+bx+c)+r⇔x3_{+10 =ax}3_+(2a+b)x2_{+(2b+c)x+(2c+r)✳}

▲♦❣♦✱ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ t❡♠♦s

a= 1✱

2a+b = 0_⇒b=₋2a_⇒b=₋2✱

2b+c= 0_⇒c=₋2b _⇒c= 4✱

2c+r= 10_⇒r= 10₋2c_⇒r= 2✳

P♦rt❛♥t♦✱ q(x) =x2_{−2x+ 4 ❡ r(x) = 2✳ ⋄}

❖✉tr❛ s♦❧✉çã♦✿ P❛rt✐♥❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡

x3+ 8 = (x+ 2)(x2₋2x+ 4)

❡ s♦♠❛♥❞♦ ✷ ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s✱ t❡♠✲s❡

x3_{+ 8 + 2 =x}3 _{+ 10 = (x+ 2)(x}2_{−2x+ 4) + 2.}

▲♦❣♦✱ q(x) =x2_{−2x+ 4 ❡ r(x) = 2✳}

❖❜s❡r✈❛çã♦✿ ❊①✐st❡ t❛♠❜é♠ ✉♠ ♠ét♦❞♦ ❞❡ ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ♠✉✐t♦ ✉s❛❞♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❝❤❛♠❛❞♦ ♠ét♦❞♦ ❞❛s ❝❤❛✈❡s✳

❚❡♦r❡♠❛ ✶✳✾ ✭ ❚❡♦r❡♠❛ ❞❡ ❉✬❆❧❡♠❜❡rt✮ ❖ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦

p(x) ♣♦r x₋a é p(a)✳

❉❡♠♦♥str❛çã♦✿ ❉❛ ❞✐✈✐sã♦ ❞❡ p(x) ♣♦r x₋a r❡s✉❧t❛ ✉♠ q✉♦❝✐❡♥t❡ q(x) ❡ ✉♠

r❡st♦ r(x) t❛✐s q✉❡ p(x) = (x₋a)q(x) +r(x)✳ ❈♦♠♦ ♦ ❞✐✈✐s♦r x₋a é ❞❡ ❣r❛✉ ✶✱

♦ r❡st♦ s❡rá ❞❡ ❣r❛✉ ③❡r♦✱ ♦✉ s❡❥❛✱ ✉♠❛ ❝♦♥st❛♥t❡✳ ❋❛③❡♥❞♦ r(x) = r✱ ❝♦♥st❛♥t❡✱

t❡♠♦s✿ p(x) = (x₋a)q(x) +r ❡ s✉❜st✐t✉✐♥❞♦ x ♣♦r a s❡❣✉❡✲s❡ q✉❡

p(a) = (a₋a)q(a) +r_⇒r =p(a).

❊①❡♠♣❧♦s✿

✶✳ ❈❛❧❝✉❧❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡p(x) = 2x3_{+ 3x}2 _{+ 2x−5♣♦r x−1✳}

❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❉✁❆❧❡♠❜❡rt✿

r =p(1) = 2(1)3+ 3(1)2+ 2(1)₋5 = 2.

✷✳ ❊♥❝♦♥tr❡ ♦ r❡st♦ ❞❡ ❞✐✈✐sã♦ ❞❡ p(x) = x4_{+ 2x}2_{+ 2x−3♣♦r x+ 2✳}

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❉✁❆❧❡♠❜❡rt t❡♠♦s✿

r =p(₋2) = (₋2)4 + 2(₋2)2+ 2(₋2)₋3 = 17.

⋄

❈♦r♦❧ár✐♦ ✶✳✾✳✶ ✭❚❡♦r❡♠❛ ❞♦ ❢❛t♦r✮ ❯♠ ♣♦❧✐♥ô♠✐♦ p(x) é ❞✐✈✐sí✈❡❧ ♣♦rx₋as❡✱

❡ s♦♠❡♥t❡ s❡✱ p(a) = 0✱ ♦✉ s❡❥❛✱ p(x) = (x₋a)q(x)✳

✶✳✶✳ P❖▲■◆Ô▼■❖ ❊▼ ❯▼❆ ❱❆❘■➪❱❊▲

❉❡♠♦♥str❛çã♦✿ ❙❡p(x)é ❞✐✈✐sí✈❡❧ ♣♦rx₋a✱ ❡♥tã♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❉✬❆❧❡♠❜❡rt✱ r =p(a) = 0✱ ❡✱ ❞❡ ♦✉tr❛ ❢♦r♠❛✱ s❡ p(a) = 0✱ ❝♦♠♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❉✬❆❧❡♠❜❡rt✱

r =p(a)✱ t❡♠♦s r= 0✱ ♦✉ s❡❥❛✱ p(x)é ❞✐✈✐sí✈❡❧ ♣♦r x₋a✳

❊①❡♠♣❧♦✿ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ k ❞❡ ♠♦❞♦ q✉❡ p(x) = x3 _+x2 _{+kx−2 s❡❥❛}

❞✐✈✐sí✈❡❧ ♣♦r x+1 2✳

❙♦❧✉çã♦✿ ❉❡✈❡♠♦s t❡r p

−1₂

= 0✳

▲♦❣♦✿

−1₂

3

+

−1₂

2

+k

−1₂

−2 = 0

⇒ −1

8 + 1 4 −

k

2 −2 = 0⇒

k

2 =− 15

8 ⇒k =− 15

4✳ ⋄

✶✳✶✳✺ ❉✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐

❚❡♦r❡♠❛ ✶✳✶✵ ❆♦ ❞✐✈✐❞✐r ♦ ♣♦❧✐♥ô♠✐♦ p(x) = anxn+an−1xn−1+...+a1x+a0 ♣♦r

x₋a ♦❜t❡♠♦s ✉♠ q✉♦❝✐❡♥t❡q(x) =bn−1xn−1+· · ·+b1x+b0 ❡ r❡st♦ r✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ♦ ❣r❛✉ ❞❡ p(x)é n ❡ ♦ ❣r❛✉ ❞❡x₋a é ✶✱ ♦ ❣r❛✉ ❞❡q(x)é

♥♦ ♠á①✐♠♦ n₋1✳ ❆ss✐♠✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ t❡♠♦s✿

p(x) =anxn+an−1xn−1+· · ·+a1x+a0 = (x−a)(bn−1xn−1+· · ·+b1x+b0) +r.

❉❡s❡♥✈♦❧✈❡♥❞♦ ♦ s❡❣✉♥❞♦ ♠❡♠❜r♦✱ ♦❜t❡♠♦s✱ ♣♦r ❝♦♠♣❛r❛çã♦✿

bn−1 =an

bn−2 =abn−1+an−1 =aan+an−1

✳✳✳ ✳✳✳ ✳✳✳

b1 =ab2+a2

b0 =ab1+a1

❈♦♠ ❝♦❡✜❝✐❡♥t❡s ❞❛❞♦s ❞❛ ❢♦r♠❛ ❛❝✐♠❛✱ t❡♠♦s✿

q(x) =bn−1xn−1+· · ·+b1x+b0 ❡ r=ab0+a0

❖❜s❡r✈❛çã♦✿ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ❡❢❡t✉❛❞♦ ❛❝✐♠❛ ♣♦❞❡ s❡r ❝♦❧♦❝❛❞♦ ♥❛ s❡❣✉✐♥t❡ ❢♦r♠❛

♣rát✐❝❛ ♥♦ ♠♦❞♦ ❤♦r✐③♦♥t❛❧✳ ❖❜s❡r✈❡✿

a an an−1 an−2 · · · a2 a1 a0

bn−1 bn−2 bn−3 · · · b1 b0 r

❊①❡♠♣❧♦✿ ❯s❡ ♦ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐ ♣❛r❛ ♦❜t❡r ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ p(x) = 3x5_{+ 4x}4_{+ 3x}3_−7x2_{−2x+ 3 ♣♦r d(x) =x−1✳}

❙♦❧✉çã♦✿ ❈♦♥❢♦r♠❡ ♦ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐✱ t❡♠♦s✿

x₋1 = 0_⇒x= 1_⇒a= 1 ❡ grau(q) =grau(p)₋grau(d) = 5₋1 = 4

❡s❝r❡✈❡♥❞♦ q(x) = q4x4+q3x3+q2x2+q1x+q0 ❡r =aq0+a0✳

❈♦♠♦✿ a5 = 3, a4 = 4, a3 = 3, a2 =−7, a1 =−2 ❡a0 = 3✱ t❡♠♦s✿

q4 =a5 = 3

q3 =aq4+a4 = 1·3 + 4 = 7

q2 =aq3+a3 = 1·7 + 3 = 10

q1 =aq2+a2 = 1·10 + (−7) = 3

q0 =aq1+a1 = 1·3 + (−2) = 1 ❡r =aq0+a0 = 1·1 + 3 = 4

P♦rt❛♥t♦✱ q(x) = 3x4_{+ 7x}3 _{+ 10x}2_{+ 3x+ 1 ❡ r = 4. ⋄}

❖❜s❡r✈❛çã♦✿ ❖ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐ é ♠✉✐t♦ út✐❧ q✉❛♥❞♦ s❡ ❝♦♥❤❡❝❡ ✉♠❛ r❛✐③ ✐♥t❡✐r❛ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦✱ ♣♦✐s ❡ss❡ ❞✐s♣♦s✐t✐✈♦ r❡❞✉③ ♦ ❣r❛✉ ❞❛ ❡q✉❛çã♦ ❞❡ n

♣❛r❛ n₋1✳

❊①❡♠♣❧♦✿ ❘❡s♦❧✈❛ ❛ ❡q✉❛çã♦ x3_−3x2_{+ 4x−2 = 0✳}

◆♦t❡ q✉❡ ✶ é r❛✐③ ❞❛ ❡q✉❛çã♦ ❞♦ ✸♦ _{❣r❛✉ x}3 _{− 3x}2 _{+ 4x−2 = 0✳ ❯s❛♥❞♦ ♦}

❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐✿

1 1 ₋3 4 ₋2 1 ₋2 2 0

❡♥❝♦♥tr❛r❡♠♦sx2_{−2x+2 = 0✱ q✉❡ é ✉♠❛ ❡q✉❛çã♦ ❞♦ ✷}♦ _{❣r❛✉ ❝♦♠ r❛í③❡s1+i❡1−i✳}

P♦rt❛♥t♦✱ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ sã♦✿ ✶✱ 1 +i ❡ 1₋i✳

✶✳✷✳ ❊◗❯❆➬Õ❊❙ P❖▲■◆❖▼■❆■❙ ❖❯ ❆▲●➱❇❘■❈❆❙

✶✳✷ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ♦✉ ❛❧❣é❜r✐❝❛s

❉❡✜♥✐çã♦ ✶✳✶✶ ❉❡♥♦♠✐♥❛✲s❡ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ♦✉ ❛❧❣é❜r✐❝❛ t♦❞❛ ❡q✉❛çã♦ q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛✿ anxn+an−1xn−1+· · ·+a1x+a0 = 0 ❝♦♠ an 6= 0 ❡

n _∈N∗✱ ❡♠ q✉❡ an, an−1, ..., a1, a0 sã♦ ♥ú♠❡r♦s r❡❛✐s ❡ n é ♦ ❣r❛✉ ❞❛ ❡q✉❛çã♦✳

❊①❡♠♣❧♦s✿

✶✳ 5x+ 2 = 0 é ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❞♦ ✶♦ ❣r❛✉✳

✷✳ x2_{−3x+ 2 = 0 é ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❞♦ ✷}♦ _{❣r❛✉✳}

✸✳ 7x3_−2x2_{−4x+ 5 = 0 é ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❞♦ ✸}♦ _{❣r❛✉✳}

⋄

✶✳✷✳✶ ❘❛✐③ ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ♦✉ ❛❧❣é❜r✐❝❛

❉❡✜♥✐çã♦ ✶✳✶✷ ❉❡♥♦♠✐♥❛✲s❡ r❛✐③ ♦✉ ③❡r♦ ❞❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛

anxn+an−1xn−1+· · ·+a1x+a0 = 0

♦ ✈❛❧♦r α q✉❡ s✉❜st✐t✉í❞♦ ♥♦ ❧✉❣❛r ❞❡ ① s❛t✐s❢❛③ ❛ ✐❣✉❛❧❞❛❞❡✱ ♦✉ s❡❥❛✱ ♦ ✈❛❧♦r α t❛❧

q✉❡

anαn+an−1αn−1+· · ·+a1α+a0 = 0.

❊①❡♠♣❧♦✿ ❆ ❡q✉❛çã♦x3_{−6x+4 = 0❛❞♠✐t❡x= 2❝♦♠♦ r❛✐③✱ ♣♦✐s(2)}3_{−6(2)+4 =}

8₋12 + 4 = 0✳ _⋄

✶✳✷✳✷ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ á❧❣❡❜r❛

❖ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ á❧❣❡❜r❛ ❢♦✐ ❞❡♠♦♥str❛❞♦ ♣♦r ●❛✉ss ❡♠ ✶✼✾✾ ❡ s❡rá ❡①♣♦st♦ s❡♠ ❞❡♠♦♥str❛çã♦✱ ♣♦✐s r❡q✉❡r ✉♠ ♣♦✉❝♦ ❞❡ ❛♥á❧✐s❡ ♠❛t❡♠át✐❝❛✳

❚❡♦r❡♠❛ ✶✳✶✸ ❚♦❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❣r❛✉ n (n _≥ 1) ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠✲

♣❧❡①♦s ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ r❛✐③ ❝♦♠♣❧❡①❛✳

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ á❧❣❡❜r❛ t❡♠✲s❡✿

❈♦r♦❧ár✐♦ ✶✳✶✸✳✶ ✭❚❡♦r❡♠❛ ❞❛ ❞❡❝♦♠♣♦s✐çã♦✮ ❚♦❞♦ ♣♦❧✐♥ô♠✐♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s p(x) = anxn+an−1xn−1+· · ·+a1x+a0 = 0 ❝♦♠ n ≥ 1 ❡ an 6= 0 ♣♦❞❡

s❡r ❞❡❝♦♠♣♦st♦ ♥✉♠ ♣r♦❞✉t♦ ❞❡ n ❢❛t♦r❡s ❞❡ ✶♦ ❣r❛✉✱ ♦✉ s❡❥❛✿

p(x) = an(x−x1)(x−x2)(x−x3)· · ·(x−xn).

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ p(x) ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ n (n _≥ 1)✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❢✉♥✲

❞❛♠❡♥t❛❧ ❞❛ á❧❣❡❜r❛ p(x)❛❞♠✐t❡ x1 ∈C ❝♦♠♦ r❛✐③✱ ♦✉ s❡❥❛✱p(x1) = 0✱ ❡♥tã♦ ❡①✐st❡

✉♠ q(x)✭q✉♦❝✐❡♥t❡✮ ❞❡ ❣r❛✉ n₋1 t❛❧ q✉❡✿

p(x) = (x₋x1)q(x).

❆❣♦r❛ s❡ q(x) t✐✈❡r ❣r❛✉ n₋1_≥ 1✱ ❞❡ ♥♦✈♦ ❡①✐st❡♠ x2 ∈C ❡ q2(x) ❞❡ ❣r❛✉ n−2

t❛❧ q✉❡✿

q1(x) = (x−x2)q2(x)⇒p(x) = (x−x1)(x−x2)q2(x).

P♦r s✉❛ ✈❡③✱ s❡ q2(x) t✐✈❡r ❣r❛✉ n−2≥1❡①✐st❡♠ x3 ∈C ❡q3(x) ❞❡ ❣r❛✉n−3 t❛❧

q✉❡✿

q2(x) = (x−x3)q3(x)⇒p(x) = (x−x1)(x−x2)(x−x3)q3(x).

❆♣❧✐❝❛♥❞♦ ♦ ♣r♦❝❡ss♦ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❝♦♥❝❧✉í♠♦s q✉❡

p(x) = an(x−x1)(x−x2)(x−x3)· · ·(x−xn).

❆ ♠❡♥♦s ❞❡ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✱ ❛ ❞❡❝♦♠♣♦s✐çã♦ é ú♥✐❝❛✳

❈♦r♦❧ár✐♦ ✶✳✶✸✳✷ ❚♦❞❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❞❡ ❣r❛✉n❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ♣♦s✲

s✉✐ ❡①❛t❛♠❡♥t❡ n r❛í③❡s ❝♦♠♣❧❡①❛s✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✶✸✳✶ t♦❞♦ ♣♦❧✐♥ô♠✐♦ p(x) ❞❡ ❣r❛✉ n ♣♦❞❡ s❡r

❞❡❝♦♠♣♦st♦ ♥✉♠ ♣r♦❞✉t♦ ❞❡ n ❢❛t♦r❡s ❞♦ ✶♦ ❣r❛✉✱ ❧♦❣♦✿

p(x) = 0_⇒an(x−x1)(x−x2)(x−x3)· · ·(x−xn) = 0,

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❛ ❡q✉❛çã♦ p(x) = 0 ♣♦ss✉✐ ❡①❛t❛♠❡♥t❡ n r❛í③❡s ❝♦♠♣❧❡①❛s✳

❊①❡♠♣❧♦✿ ❱❛♠♦s ❢❛③❡r ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞♦ ♣♦❧✐♥ô♠✐♦ p(x) =x5_{−16x❡♠ ❢❛t♦r❡s}

❞❡ ✶♦ _{❣r❛✉✿}

❙♦❧✉çã♦✿ ❱❛♠♦s ❞❡❝♦♠♣♦r p(x) = x5_{−16x ♣❡❧♦ ♠ét♦❞♦ ❞❛ ❢❛t♦r❛çã♦❀ ❧♦❣♦}

p(x) =x5₋16x=x(x4₋16) =x(x2₋4)(x2+ 4) = x(x₋2)(x+ 2)(x₋2i)(x+ 2i).

P♦rt❛♥t♦✱ p(x) =x(x₋2)(x+ 2)(x₋2i)(x+ 2i) é ❞♦ ✺♦ ❣r❛✉ ❡ ♣♦ss✉✐ ❡①❛t❛♠❡♥t❡

✺ r❛í③❡s ❝♦♠♣❧❡①❛s✳ ⋄

▼✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ ✉♠❛ r❛✐③

❉❡✜♥✐çã♦ ✶✳✶✹ ❙❡ ✉♠ ♣♦❧✐♥ô♠✐♦ p(x) é t❛❧ q✉❡✿

p(x) = (x₋α)m_·q(x)

❝♦♠ q(α)₆= 0✱ ❞✐③❡♠♦s q✉❡ α é r❛✐③ ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ♠ ❞❛ ❡q✉❛çã♦ p(x) = 0✳