UM ESTUDO SOBRE EQUAÇÕES POLINOMIAIS DEDICADO AO ENSINO BÁSICO

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❯♠ ❡st✉❞♦ s♦❜r❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s

❞❡❞✐❝❛❞♦ ❛♦ ❊♥s✐♥♦ ❇ás✐❝♦

P❛trí❝✐❛ ❈❛s❛r♦t♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

P❛trí❝✐❛ ❈❛s❛r♦t♦

❯♠ ❡st✉❞♦ s♦❜r❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡❞✐❝❛❞♦ ❛♦ ❊♥s✐♥♦

❇ás✐❝♦

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦ ❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r✳ ❊✈❡r❛❧❞♦ ❞❡ ▼❡❧❧♦ ❇♦♥♦tt♦ ■❈▼❈✴❯❙P ✲ ❙ã♦ ❈❛r❧♦s ✲ ❙P

Pr♦❢❛✳ ❉r❛✳ ❊❧✐r✐s ❈r✐st✐♥❛ ❘✐③③✐♦❧❧✐ ■●❈❊✴ ❯◆❊❙P ✲ ❘✐♦ ❈❧❛r♦ ✲ ❙P

❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ❉❡✉s ♣♦r ♠❡ ❛❝♦♠♣❛♥❤❛r ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s ♠❡✉s ♣❛✐s✱ ❏♦ã♦ ❇❛t✐st❛ ❡ ▼❛r✐❛ ❍❡❧❡♥❛ q✉❡✱ ❛tr❛✈és ❞♦s s❡✉s ❡s❢♦rç♦s✱ ♠❡ ❞❡r❛♠ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❡st✉❞♦s✱ ♣❡r♠✐t✐♥❞♦ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ s♦♥❤♦✳

➚s ♠✐♥❤❛s ✐r♠ãs✱ ❆❧❡ss❛♥❞r❛ ❡ ❈r✐st✐❛♥❡✱ ♣❡❧♦s ❝♦♥s❡❧❤♦s ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✳ ❆♦ ♠❡✉ ♠❛r✐❞♦✱ ❍❡r❛❧❞♦✱ ♦ q✉❛❧ ❞❡❞✐❝♦✉ ♦s s❡✉s ✜♥s ❞❡ s❡♠❛♥❛ ❛ ♠❡ ❛❝♦♠♣❛♥❤❛r ♥❡ss❛ ❥♦r♥❛❞❛ ❝♦♠ ♠✉✐t❛ ♣❛❝✐ê♥❝✐❛✱ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦✳

➚ ♠✐♥❤❛ ✜❧❤❛✱ ❆❧✐❝❡✱ ❡ ♠❡✉ s♦❜r✐♥❤♦✱ ❏♦ã♦ ❱✐❝t♦r✱ q✉❡ t♦r♥❛♠ ♦s ♠❡✉s ❞✐❛s ♠❛✐s ❢❡❧✐③❡s✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ P❘❖❋▼❆❚✱ ♣❡❧❛ tr♦❝❛ ❞❡ ❡①♣❡r✐ê♥❝✐❛s ❡ ❛ ❛♠✐③❛❞❡✳ ➚ ❈❆P❊❙ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳

❆♦s ♠❡✉s ❛❧✉♥♦s✱ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ♥❛ ❜✉s❝❛ ♣♦r ✉♠ ❡♥s✐♥♦ ♠❡❧❤♦r✳

❘❡s✉♠♦

❆ ❣r❛♥❞❡ ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❡♥s✐♥❛r ▼❛t❡♠át✐❝❛ ❛♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ s❡ ❞á ♣❡❧❛ ❢❛❧t❛ ❞❡ ✐♥t❡r❡ss❡ ❞♦s ❛❧✉♥♦s ❝♦♠ ♦s ❝♦♥t❡ú❞♦s✱ q✉❡ ♠✉✐t❛s ✈❡③❡s sã♦ ❡♥s✐♥❛❞♦s ❞❡ ❢♦r♠❛ ❞❡s❝♦♥t❡①t✉❛❧✐③❛❞❛✳ ◆❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ♣♦❧✐♥ô♠✐♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ♥❛ ❢♦r♠❛ ❞❡ ❛❧❣♦r✐t♠♦s✱ ✈✐s❛♥❞♦ ❛ ✜①❛çã♦ ♥❛ ❢♦r♠❛ ❞❡ r❡♣❡t✐çã♦ s❡♠ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ s✐t✉❛çã♦ ❞♦ ❞✐❛✲❛✲❞✐❛ ♣❛r❛ ✐❧✉str❛r ♦ ♣r♦❜❧❡♠❛✳ ❉✐❛♥t❡ ❞❡ss❛ r❡❛❧✐❞❛❞❡✱ ♣r❡t❡♥❞❡♠♦s ❡st✐♠✉❧❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❡ ✐♥❝❡♥t✐✈❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ♣♦❧✐♥ô♠✐♦s ❡ s♦❜r❡ ❛s té❝♥✐❝❛s ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐✲ ❛✐s✳ ❆ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ❝♦♥t❡♠♣❧❛ ✉♠ ♣❧❛♥♦ ❞❡ ❛✉❧❛ q✉❡ r❡❧❛❝✐♦♥❛ ♦s ❝♦♥t❡ú❞♦s ❝♦♠ ❋ís✐❝❛✱ ❊❝♦♥♦♠✐❛ ❡ ❆❞♠✐♥✐str❛çã♦✳

❆❜str❛❝t

❚❤❡ ❞✐✣❝✉❧t② ♦❢ t❡❛❝❤✐♥❣ ▼❛t❤❡♠❛t✐❝s t♦ st✉❞❡♥ts ♦❢ ❇❛s✐❝ ❊❞✉❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❧❛❝❦ ♦❢ ✐♥t❡r❡st ♦❢ t❤❡ st✉❞❡♥ts ✇✐t❤ t❤❡ ❝♦♥t❡♥ts✱ ✇❤✐❝❤ ❛r❡ ♦❢t❡♥ t❛✉❣❤t ✐♥ ❛ ❞❡❝♦♥✲ t❡①t✉❛❧✐③❡❞ ✇❛②✳ ■♥ ♠❛♥② ❜♦♦❦s✱ t❤❡ ❝♦♥❝❡♣ts r❡❧❛t❡❞ t♦ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛❧❣♦r✐t❤♠s✱ ✇✐t❤♦✉t ❞❡✈❡❧♦♣✐♥❣ ❛ s✐t✉❛t✐♦♥ ♦❢ t❤❡ ❞❛✐❧② r♦✉t✐♥❡ t♦ ✐❧❧✉str❛t❡ t❤❡ ♣r♦❜❧❡♠✳ ●✐✈❡♥ t❤✐s r❡❛❧✐t②✱ ✇❡ ✇❛♥t t♦ st✐♠✉❧❛t❡ t❤❡ ❝✉r✐♦s✐t② ❛♥❞ ❡♥❝♦✉r❛❣❡ t❤❡ ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ ❜❛s✐❝s ♦❢ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ♦♥ t❡❝❤♥✐q✉❡s ❢♦r s♦❧✈✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s✳ ❚❤❡ ♣r♦♣♦s❛❧ ❝♦♠♣r✐s❡s ❛ ❞✐❞❛❝t✐❝ ❧❡ss♦♥ ♣❧❛♥ t❤❛t ❧✐sts t❤❡ ❝♦♥t❡♥ts ✇✐t❤ P❤②s✐❝s✱ ❊❝♦♥♦♠✐❝s ❛♥❞ ❆❞♠✐♥✐str❛t✐♦♥✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶✼

◆♦t❛s ❤✐stór✐❝❛s s♦❜r❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✶✾

✶ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ✷✸

✶✳✶ P♦❧✐♥ô♠✐♦s ❡ ♦♣❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✶✳✶ ❆❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✶✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✷ ❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✸ ❘❛✐③ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✹ ❉✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✷ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✹✺

✷✳✶ ❘❡❧❛çõ❡s ❡♥tr❡ ❝♦❡✜❝✐❡♥t❡s ❡ r❛í③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✷ ❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✸ Pr♦♣♦st❛s ❞❡ ❛t✐✈✐❞❛❞❡s ❞✐❞át✐❝❛s ✺✾

✸✳✶ ❋✐♥❛♥❝✐❛♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✷ Pr♦❜❧❡♠❛s ❞❡ ❖t✐♠✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✸ ▼❛t❡♠á❣✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

■♥tr♦❞✉çã♦

◆♦ ❈❛♣ít✉❧♦ ❱■■■ ❞♦ ❘❡❣✐♠❡♥t♦ ❞♦ Pr♦❢▼❛t✱ ❝♦♥st❛✿

✧❆rt✐❣♦ ✷✽ ✲ ❖ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❞❡✈❡ ✈❡rs❛r s♦❜r❡ t❡♠❛s ❡s♣❡❝í✜❝♦s ♣❡rt✐♥❡♥t❡s ❛♦ ❝✉rrí❝✉❧♦ ❞❡ ▼❛t❡♠át✐❝❛ ❊♥s✐♥♦ ❇ás✐❝♦ ❡ q✉❡ t❡♥❤❛♠ ✐♠♣❛❝t♦ ♥❛ ♣rát✐❝❛ ❞✐❞át✐❝❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✧✳

❉❡♥tr♦ ❞❡ss❛s ❞✐r❡tr✐③❡s✱ ♦ tó♣✐❝♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♥❡st❡ tr❛❜❛❧❤♦ ❢♦✐✿ ❊q✉❛çõ❡s ♣♦✲ ❧✐♥♦♠✐❛✐s✳

❖ ❝♦♥t❡ú❞♦ ❡s♣❡❝í✜❝♦ ❞♦ tr❛❜❛❧❤♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ❛ ♣❛rt✐r ❞❡ ❝♦♥❝❡✐t♦s✱ ♣r♦♣r✐❡✲ ❞❛❞❡s ❡ r❡s✉❧t❛❞♦s ❡ss❡♥❝✐❛✐s s♦❜r❡ ♣♦❧✐♥ô♠✐♦s ❛té ❝❤❡❣❛r ♥❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞♦✱ t❡r❝❡✐r♦ ❡ q✉❛rt♦ ❣r❛✉✳ ❖s ♠ét♦❞♦s ❛♣r❡s❡♥t❛❞♦s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ t❡r❝❡✐r♦ ❡ q✉❛rt♦ ❣r❛✉ ♥ã♦ ❝♦st✉♠❛♠ s❡r ❛❜♦r❞❛❞❛s ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ♠❛s ❛s ❢❡rr❛♠❡♥t❛s sã♦ ❝♦♥❤❡❝✐❞❛s ♣❡❧♦s ❛❧✉♥♦s✱ s❡♥❞♦ ❡ss❛ ❛ ❥✉st✐✜❝❛t✐✈❛ ♣♦r ✐♥❝❧✉✐r t❛✐s ♠ét♦❞♦s ♥♦ tr❛❜❛❧❤♦✳

P❛r❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝♦♥t❡ú❞♦✱ ♣r♦♣♦♠♦s✱ ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ ✉♠ ♣❧❛♥♦ ❞❡ ❛✉❧❛ ❝♦♠ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ ❋ís✐❝❛✱ ❆❞♠✐♥✐str❛çã♦ ❡ ❊❝♦♥♦♠✐❛✳ ❆ ✜♠ ❞❡ ❞✐✈❡rs✐✜✲ ❝❛r ❡ ❞✐♥❛♠✐③❛r ❛s ❛✉❧❛s✱ s✉❣❡r✐♠♦s ♦ ✉s♦ ❞❛ ❝❛❧❝✉❧❛❞♦r❛ ❡ ❞♦ r❡❝✉rs♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❲✐♥♣❧♦t✳

❆ ✜♥❛❧✐❞❛❞❡ ❞❡ss❡ ♠❛t❡r✐❛❧ é ❡st✐♠✉❧❛r ❡ ❛✉①✐❧✐❛r ♦ ❛❧✉♥♦ ❛ ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ♦ t❡♠❛ ❞❡s❡♥✈♦❧✈✐❞♦✳ P❛rt✐♥❞♦ ❞♦ ♣r✐♥❝í♣✐♦ q✉❡ ❛ ▼❛t❡♠át✐❝❛ ❧❡✈♦✉ ♠✐❧ê♥✐♦s ♣❛r❛ s❡r ❝♦♥str✉í❞❛✱ ❡s♣❡r❛✲s❡ q✉❡ ♦ ❧❡✐t♦r✲❛❧✉♥♦ ♥ã♦ ❞❡s✐st❛ ❞❡ ❛♣r❡♥❞ê✲❧❛✱ s❡ ❡♥❝♦♥tr❛r ❞✐✜❝✉❧❞❛❞❡s✳

■♥❢♦r♠❛♠♦s q✉❡✱ ♠❡s♠♦ q✉❡ ♥❡♠ t♦❞❛s t❡♥❤❛♠ s✐❞♦ ❝✐t❛❞❛s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ t♦❞❛s ❛s r❡❢❡rê♥❝✐❛s ♣r❡s❡♥t❡s ❛♦ ✜♥❛❧ ❞♦ tr❛❜❛❧❤♦ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ♠❡s♠♦✳

◆♦t❛s ❤✐stór✐❝❛s s♦❜r❡ ❡q✉❛çõ❡s

♣♦❧✐♥♦♠✐❛✐s

➚ ♠❡❞✐❞❛ q✉❡ ♦ ❤♦♠❡♠ ❝♦♠❡ç♦✉ ❛ ❝❛❧❝✉❧❛r✱ ❝♦♥t❛♥❞♦ r❡❜❛♥❤♦s✱ tr♦❝❛♥❞♦ ♣r♦❞✉t♦s✱ ❝♦♥t❛❜✐❧✐③❛♥❞♦ ✐♠♣♦st♦s ♦✉ ❝♦♥str✉✐♥❞♦ ♦s ♣r✐♠❡✐r♦s ♠♦♥✉♠❡♥t♦s ❡ ♦❜r❛s ❞❡ ❡♥❣❡♥❤❛✲ r✐❛✱ ❛s ❢♦r♠❛s ♠❛✐s s✐♠♣❧❡s ❞❛s ❝❤❛♠❛❞❛s ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ❛♣r❡s❡♥t❛r❛♠✲s❡ q✉❛s❡ q✉❡ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❛♦s ❛♥t✐❣♦s ♠❛t❡♠át✐❝♦s✳

❊q✉❛çõ❡s ❛❧❣é❜r✐❝❛s sã♦ ❛q✉❡❧❛s ❡♠ q✉❡ ❛ ✐♥❝ó❣♥✐t❛ ❛♣❛r❡❝❡ ❛♣❡♥❛s s✉❜♠❡t✐❞❛ às ❝❤❛♠❛❞❛s ♦♣❡r❛çõ❡s ❛❧❣é❜r✐❝❛s✱ ❛ s❛❜❡r s♦♠❛ ♦✉ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞✐✈✐sã♦✱ ♣♦t❡♥❝✐❛çã♦ ✐♥t❡✐r❛ ✭q✉❡ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ n❢❛t♦r❡s

✐❣✉❛✐s✮ ❡ r❛❞✐❝✐❛çã♦✳ ❈♦♠♦ ❡①❡♠♣❧♦s ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s✱ t❡♠♦s✿

ax+b =c,

ax2+bx+c= 0,

x5+√4x5_{+ 9 = 10x,}

x4 + 3x−2 ₌√3

x5_{+ 14.}

P♦r ♦✉tr♦ ❧❛❞♦✱

x2+ 5x+ 3 =e−x_,

cosx+x2cos 2x= 8,

arctgx= π 4,

♥ã♦ sã♦ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s✳

◗✉❛♥❞♦ ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ é ❝♦❧♦❝❛❞❛ s♦❜ ❛ ❢♦r♠❛

a₀+a₁x+_{· · ·}+an−1xn−1+anxn = 0 (n ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦), ❞✐③✲s❡ q✉❡ ❡❧❛ ❡stá ❡♠ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❡ é ❞❡♥♦♠✐♥❛❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧✳

❖ ♠❛✐♦r ❡①♣♦❡♥t❡ ❞❛ ✐♥❝ó❣♥✐t❛x❡♠ ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❡♠ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛

é ❞❡♥♦♠✐♥❛❞♦ ❣r❛✉ ❞❛ r❡❢❡r✐❞❛ ❡q✉❛çã♦✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦ ❈❛♣ít✉❧♦✷✳

❊♠❜♦r❛ ❛s ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ❥á t❡♥❤❛♠ ♠❡r❡❝✐❞♦ ❛ ❛t❡♥çã♦ ❞♦s ❡❣í♣❝✐♦s✱ ❝❡r❝❛ ❞❡ ✷✵✵✵ ❛♥♦s ❛✳❈✳ ♥❛ ❜✉s❝❛ ❞❡ s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ♣rát✐❝♦s✱ ❝♦♠♦ ♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛ ❞✐✈✐sã♦ ❞❡ t❡rr❛s ❡ ❤❡r❛♥ç❛s✱ ❢♦r❛♠ ♦s ❡st✉❞♦s ♣✉r❛♠❡♥t❡ t❡ór✐❝♦s r❡❛❧✐③❛❞♦s ♣❡❧♦s

✷✵

❣r❡❣♦s✱ ❝❡r❝❛ ❞❡ ✸✵✵ ❛♥♦s ❛✳❈✳✱ q✉❡ ❝r✐❛r❛♠ ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ s❡ ❡♥❝♦♥tr❛ss❡ ✉♠ ♠ét♦❞♦ ❣❡r❛❧ ♣❛r❛ r❡s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡1◦ ❣r❛✉✳ ❚❛❧ ♠ét♦❞♦ ❢♦✐ ❞❡❞✉③✐❞♦ ❛ ♣❛rt✐r

❞♦s ♣♦st✉❧❛❞♦s ❡♥✉♥❝✐❛❞♦s ♥❛ ❝♦♥❤❡❝✐❞❛ ♦❜r❛ ❖s ❊❧❡♠❡♥t♦s✱ ❞❡ ❊✉❝❧✐❞❡s✳

❆ ❢ór♠✉❧❛ q✉❡ ❝♦♥❤❡❝❡♠♦s ❝♦♠♦ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✱ ♥❛ ✈❡r❞❛❞❡✱ ♥ã♦ ❢♦✐ ❞❡s❝♦✲ ❜❡rt❛ ♣♦r ❇❤❛s❦❛r❛ ✭✶✶✶✹✲✶✶✽✺✮✱ ❡❧❛ ❢♦✐ ♣✉❜❧✐❝❛❞❛ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❙r✐❞❤❛r❛ ✉♠ sé❝✉❧♦ ❛♥t❡s ❞❡ ❇❤❛s❦❛r❛ ❡♠ ✉♠❛ ♦❜r❛ q✉❡ ♥ã♦ ❝❤❡❣♦✉ ❛té ♥ós✳

❋♦✐ ❛ ♣❛rt✐r ❞❛ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ q✉❡ s✉r❣✐r❛♠ ❞✉❛s ❝✉r✐♦s✐❞❛❞❡s ✐♠♣♦rt❛♥t❡s✿ ✐✳ ❊q✉❛çõ❡s ❞❡ ❣r❛✉ ♠❛✐♦r ❞♦ q✉❡ ✶ ♣♦❞❡r✐❛♠ t❡r ♠❛✐s ❞❡ ✉♠❛ s♦❧✉çã♦❀

✐✐✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ ❛ ❢ór♠✉❧❛ ❢♦r♥❡❝✐❛ ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦✱ ♦ q✉❡ ♥❛ é♣♦❝❛ ♥ã♦ ❢❛③✐❛ s❡♥t✐❞♦✱ ♣♦✐s ❛✐♥❞❛ ♥ã♦ s❡ ❝♦♥❤❡❝✐❛♠ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ◆❡ss❡s ❝❛s♦s✱ t❛✐s s✐t✉❛çõ❡s ❡r❛♠ ✐♥t❡r♣r❡t❛❞❛s ❝♦♠♦ ❛ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦✳

❈♦♥❢♦r♠❡ r❡❣✐str♦s ❞❡ ✶✺✶✵✱ ❙❝✐♣✐♦♥❡ ❞❡❧ ❋❡rr♦✱ ✉♠ ♠❛t❡♠át✐❝♦ ✐t❛❧✐❛♥♦✱ ❡♥❝♦♥tr♦✉ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ r❡s♦❧✈❡r ❛s ❡q✉❛çõ❡s ❞❡ 3◦ ❣r❛✉✱ ♠❛s ❡❧❡ ♠♦rr❡✉ ❛♥t❡s q✉❡ ♣✉❞❡ss❡

♣✉❜❧✐❝❛r s✉❛ ❞❡s❝♦❜❡rt❛✳ P♦ré♠✱ ❡❧❡ ❛ r❡✈❡❧♦✉ ❛♦ s❡✉ ❛❧✉♥♦ ❆♥tô♥✐♦ ▼❛r✐❛ ❋✐♦r q✉❡✱ ♣♦r s✉❛ ✈❡③✱ t❡♥t♦✉ s❡ ❛♣r♦♣r✐❛r ❞♦ ♠ér✐t♦ ❞♦ s❡✉ ♠❡str❡✳

❙❡♥❞♦ ❢r❡q✉❡♥t❡ ♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ❞❡s❛✜♦s ❡♥tr❡ ♦s sá❜✐♦s ♥❛q✉❡❧❛ é♣♦❝❛✱ ❋✐♦r ❡❧❡❣❡✉ ♦ t❛❧❡♥t♦s♦ ♠❛t❡♠át✐❝♦ ✐t❛❧✐❛♥♦ ◆✐❝❝♦❧ò ❋♦♥t❛♥❛ ✭✶✺✵✵✲✶✺✺✼✮✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❛rt❛✲ ❣❧✐❛✳ ❖ ❞❡s❛✜♦ ❝♦♥s✐st✐❛ ♥❛ s♦❧✉çã♦ ❞❡ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s q✉❡ ✉♠ ❞❡✈❡r✐❛ ♣r♦♣♦r ❛♦ ♦✉tr♦ ❡ ❋✐♦r✱ ♥❛t✉r❛❧♠❡♥t❡✱ ♣r❡t❡♥❞✐❛ ❛♣r❡s❡♥t❛r q✉❡stõ❡s q✉❡ ❞❡♣❡♥❞❡ss❡♠ ❞❛✲ q✉❡❧❡ t✐♣♦ ❞❡ ❡q✉❛çã♦ ❞❡ 3◦ ❣r❛✉✱ ❞❛ q✉❛❧ ❡❧❡ ❥á ❞❡t✐♥❤❛ ❛ s♦❧✉çã♦✳ P♦ré♠✱ ❚❛rt❛❣❧✐❛✱

❝♦♠ s✉❛ ❣❡♥✐❛❧✐❞❛❞❡✱ ❛❧é♠ ❞❡ r❡s♦❧✈❡r t♦❞❛s ❛s q✉❡stõ❡s ♣r♦♣♦st❛s ♣❡❧♦ ❞❡s❧❡❛❧ ♦♣♦✲ ♥❡♥t❡✱ ❞❡s❛✜♦✉✲♦ ❛ ❛♣r❡s❡♥t❛r ❛ s♦❧✉çã♦ ❣❡r❛❧ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞♦ 3◦ ❣r❛✉ ❞♦ t✐♣♦ x3 _+px2 _{+q = 0✳ ❋✐♦r✱ ❛♦ ❝♦♥trár✐♦ ❞❡ ❚❛rt❛❣❧✐❛✱ ♥ã♦ ❢♦✐ ❝❛♣❛③ ❞❡ ❛♣r❡s❡♥t❛r t❛❧}

s♦❧✉çã♦✳

◆❛ ♠❡s♠❛ é♣♦❝❛✱ ♦ ♠❛t❡♠át✐❝♦ ✐t❛❧✐❛♥♦ ●✐r♦❧❛♠♦ ❈❛r❞❛♥♦ ✭✶✺✵✶✲✶✺✼✻✮✱ ❡st❛✈❛ ❡s❝r❡✈❡♥❞♦ ✉♠❛ ♦❜r❛ q✉❡ ❡♥✈♦❧✈✐❛ ❝♦♥❝❡✐t♦s ❞❡ ➪❧❣❡❜r❛✱ ❆r✐t♠ét✐❝❛ ❡ ●❡♦♠❡tr✐❛✱ ❡♥tã♦ ♣r♦❝✉r♦✉ ❚❛rt❛❣❧✐❛ ❡ ♣❡❞✐✉ q✉❡ ❡❧❡ r❡✈❡❧❛ss❡ ♦ ♠ét♦❞♦ ♣❛r❛ s❡r ♣✉❜❧✐❝❛❞♦✳ ❚❛rt❛❣❧✐❛ ♥ã♦ ❛❝❡✐t♦✉ ❛ ♣r♦♣♦st❛✱ ♠❛s ❛♣ós ❥✉r❛s ❞❡ ✜❞❡❧✐❞❛❞❡✱ ❈❛r❞❛♥♦ ❝♦♥s❡❣✉✐✉ q✉❡ ❡❧❡ r❡✈❡❧❛ss❡ ♦ s❡❣r❡❞♦✳

❈❛r❞❛♥♦ tr❛✐✉ ♦s ❥✉r❛♠❡♥t♦s ❢❡✐t♦s ❛ ❚❛rt❛❣❧✐❛ ❡✱ ❡♠ ✶✺✹✺✱ ♣✉❜❧✐❝♦✉ ♥❛ ❆rs ▼❛❣♥❛ ❛ s✉❛ ❢ór♠✉❧❛✳ ❚❛rt❛❣❧✐❛ ❞❡♥✉♥❝✐♦✉ ❈❛r❞❛♥♦ ❡ ♣✉❜❧✐❝♦✉ ❛ s✉❛ ✈❡rsã♦ ❞♦s ❢❛t♦s✳ ❆♣ós tr♦❝❛r ♦❢❡♥s❛s✱ ♦ q✉❡ ♣r❡✈❛❧❡❝❡✉ ❢♦✐ ❛ ❋ór♠✉❧❛ ❞❡ ❚❛rt❛❣❧✐❛✱ ❡♠❜♦r❛ ❡❧❛ s❡❥❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❋ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✳

❆♥♦s ❞❡♣♦✐s✱ ❞❡♥tr♦ ❞♦ ❝♦st✉♠❡ ✈✐❣❡♥t❡ ❡♥tr❡ ♦s ♠❛t❡♠át✐❝♦s ❞❡ ♣r♦♣♦r❡♠ ♣r♦❜❧❡✲ ♠❛s ✉♥s ❛♦s ♦✉tr♦s✱ ✉♠ ❝❡rt♦ ❩✉❛♥♥❡ ❞❡ ❚♦♥✐♥✐ ❞❛ ❈♦✐ ♣r♦♣ôs ❛ ❈❛r❞❛♥♦ ✉♠❛ q✉❡stã♦ q✉❡ ❡♥✈♦❧✈✐❛ ❛ ❡q✉❛çã♦✿

■♥tr♦❞✉çã♦ ✷✶

❈❛r❞❛♥♦ t❡♥t♦✉ r❡s♦❧✈❡r ♠❛s ♥ã♦ ♦❜t❡✈❡ ê①✐t♦✱ ❡♥tã♦ ♣❛ss♦✉ ❛ q✉❡stã♦ ♣❛r❛ ♦ ❥♦✈❡♠ ▲✉❞♦✈✐❝♦ ❋❡rr❛r✐ ✭✶✺✷✷ ✲ ✶✺✻✵✮✱ q✉❡ ❡♥❝♦♥tr♦✉ ✉♠❛ ❢ór♠✉❧❛ ❣❡r❛❧ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡4◦

❣r❛✉✳

❆♣ós ❡ss❡s r❡s✉❧t❛❞♦s✱ ♦s ♠❛t❡♠át✐❝♦s ❝♦♠❡ç❛r❛♠ ❛ s✉s♣❡✐t❛r q✉❡ ❛s ❡q✉❛çõ❡s ❞♦

2◦ ❣r❛✉ ♣♦❞❡r✐❛♠ t❡r ❞✉❛s s♦❧✉çõ❡s ❡ ❛s ❞❡₄◦ ❣r❛✉✱ q✉❛tr♦ s♦❧✉çõ❡s ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

❋♦✐ ❡♠ ✶✼✾✾ q✉❡ ♦ ❜r✐❧❤❛♥t❡ ❛❧❡♠ã♦ ❈❛r❧ ❋r❡❞✐❝❤ ●❛✉ss ✭✶✼✼✼✲✶✽✺✺✮ ❛♣r❡s❡♥t♦✉✱ ❡♠ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦✱ ♦ ❢❛♠♦s♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛✱ ❝♦♥✜r♠❛♥❞♦ ♦ q✉❡ ❢♦r❛ s✉s♣❡✐t❛❞♦✳

❖ ❞❡s❛✜♦ ❞♦s ♠❛t❡♠át✐❝♦s ♣❛ss♦✉ ❛ s❡r ❜✉s❝❛r ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛✲ çõ❡s ❞❡ ❣r❛✉ ✺✳ ▼✉✐t❛s ❢♦r❛♠ ❛s t❡♥t❛t✐✈❛s ❞♦ ♠❛t❡♠át✐❝♦ ◆♦r✉❡❣✉ês ◆✐❡❧s ❍❡♥r✐❦ ❆❜❡❧ ✭✶✽✵✷✲ ✶✽✷✾✮✱ ♠❛s✱ ❡♠ ✶✽✷✸✱ ❡❧❡ ❞❡♠♦♥str♦✉ q✉❡✱ ❡①❝❡t♦ ❡♠ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ é ✐♠♣♦ssí✈❡❧ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ 5◦ ❣r❛✉ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s ♦♣❡r❛çõ❡s ❛❧❣é❜r✐❝❛s✳

✶ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s

✶✳✶ P♦❧✐♥ô♠✐♦s ❡ ♦♣❡r❛çõ❡s

❙❡❥❛ R ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❡ s❡❥❛ _x ✉♠ sí♠❜♦❧♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❡rt❡♥❝❡♥t❡ ❛♦ ❝♦♥❥✉♥t♦ R✱ ❞❡♥♦♠✐♥❛❞♦ ✐♥❞❡t❡r♠✐♥❛❞❛ ♦✉ ✈❛r✐á✈❡❧ s♦❜r❡ R✳

P❛r❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ j _≥ 1✱ ❞❡s✐❣♥❛r❡♠♦s ❛ j₋♣♦tê♥❝✐❛ ❞❡ x ♣♦r xj _{❡ ❡s❝r❡✲} ✈❡r❡♠♦s x1 =x✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ R ✭♦✉ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✮ é ✉♠❛ ❡①♣r❡ssã♦ ❞♦ t✐♣♦

p(x) = a0+a1x+· · ·+an−1xn−1 +anxn=✶ n X

j=0

ajxj,

♦♥❞❡ n é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❡ aj ∈R✱ ♣❛r❛ j ∈ {0,1,2,· · · , n}✳

P❛r❛j _{∈ {}0,1,2,_{· · ·}, n_}✱ ♦s ❡❧❡♠❡♥t♦saj sã♦ ❞❡♥♦♠✐♥❛❞♦s ❝♦❡✜❝✐❡♥t❡s✱ ❛s ♣❛r❝❡✲ ❧❛s ajxj sã♦ ❞❡♥♦♠✐♥❛❞❛s t❡r♠♦s ❡ ♦s t❡r♠♦s ajxj t❛✐s q✉❡ aj 6= 0 sã♦ ❞❡♥♦♠✐♥❛❞♦s ♠♦♥ô♠✐♦s ❞❡ ❣r❛✉ ❥ ❞♦ ♣♦❧✐♥ô♠✐♦ p(x)✳ ❖ ❝♦❡✜❝✐❡♥t❡ a0 é ❞❡♥♦♠✐♥❛❞♦ t❡r♠♦

❝♦♥st❛♥t❡✳

P❛r❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✱ ♦ ♣♦❧✐♥ô♠✐♦ 0(x) = 0 + 0x+ 0xn−1_{+· · ·+ 0x}n _s❡rá ❞✐t♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r 0(x) = 0✳

❯♠ ♣♦❧✐♥ô♠✐♦ s❡rá ❞✐t♦ ❝♦♥st❛♥t❡ q✉❛♥❞♦ p(x) =a0✳

■♥❢♦r♠❛♠♦s ❛♦ ❧❡✐t♦r q✉❡✱ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ ❢❛r❡♠♦s ❛s s❡❣✉✐♥t❡s ❝♦♥✈❡♥çõ❡s✿ ✶✮ ❉❡s♣r❡③❛♥❞♦ ❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✱ ❡s❝r❡✈❡r❡♠♦s ♦ ♣♦❧✐♥ô♠✐♦p(x)❝♦♠ ❛sj₋és✐♠❛s

♣♦tê♥❝✐❛s ❞❡ x ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡ ♦✉ ❡♠ ♦r❞❡♠ ❞❡❝r❡s❝❡♥t❡✱ ❛ s❛❜❡r p(x) =

a₀+a₁x+_{· · ·}+a_n−1xn−1_+a

nxn ♦✉p(x) =anxn+an−1xn−1+· · ·+a1x+a0✳

✷✮ P♦r ♥ã♦ s❡r ♥❡❝❡ssár✐♦✱ ♥ã♦ ❡s❝r❡✈❡r❡♠♦s ♦ t❡r♠♦ajxj s❡♠♣r❡ q✉❡aj = 0✱ q✉❛♥❞♦ ❤♦✉✈❡r ❛❧❣✉♠ t❡r♠♦ ♥ã♦✲♥✉❧♦ ♥♦ ♣♦❧✐♥ô♠✐♦✳

◆♦t❡ q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ p(x) = a₀+a₁x+_{· · ·}+an−1xn−1+anxn t❛♠❜é♠ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❞❛ ❢♦r♠❛ p(x) =a₀+a₁x+_{· · ·}+an−1xn−1+anxn+ 0xn+1+ 0xn+2+ 0xn+3+

✶_{▲ê✲s❡ ♦ sí♠❜♦❧♦}P❝♦♠♦ s♦♠❛tór✐❛ ♦✉ s♦♠❛ ❡ ❝♦♥✈❡♥❝✐♦♥❛✲s❡ ❡s❝r❡✈❡r_a 0x

0 =a0✳

✷✹ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s

· · ·+ 0xn+m,♦♥❞❡m é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ ✶✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱

q✉❛♥❞♦ ❝♦♠♣❛r❛r♠♦s ❞♦✐s ♣♦❧✐♥ô♠✐♦s p(x) ❡ q(x)✱ ♣♦❞❡r❡♠♦s ❛ss✉♠✐r q✉❡ ♦s t❡r♠♦s

❞❡ ❛♠❜♦s tê♠ ❛s ♠❡s♠❛s ♣♦tê♥❝✐❛s ❞❡x✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❖s ♣♦❧✐♥ô♠✐♦s p(x) = a₀ +a₁x +_{· · ·}+ an−1xn−1 +anxn ❡ q(x) =

b₀ +b₁x+_{· · ·}+bn−1xn−1 +bnxn s❡rã♦ ✐❣✉❛✐s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ aj = bj✱ ♣❛r❛ t♦❞♦

j _{∈ {}0,1,_{· · ·} , n_}✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡r❡♠♦s p(x) = q(x)✳

❖✉ s❡❥❛✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦sp(x)❡q(x)s❡ ❞❛rá ❛♣❡♥❛s q✉❛♥❞♦ t♦❞♦s

♦s ❝♦❡✜❝✐❡♥t❡ ❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ♣♦tê♥❝✐❛s ❞❡ x ❡♠ p(x) ❡ q(x) ❢♦r❡♠ ✐❣✉❛✐s✳

❊♥tã♦✱ ♦❜s❡r✈❡ q✉❡ s❡ p(x) ❡ q(x) ♥ã♦ ❢♦r❡♠ ✐❣✉❛✐s✱ ❡①✐st✐rá ❛❧❣✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧

j✱ ❝♦♠ j _{∈ {}0,1,_{· · ·} , n_}✱ t❛❧ q✉❡ aj 6= bj✳ ◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ p(x) ❡ q(x) sã♦ ❞✐❢❡r❡♥t❡s ❡ ❡s❝r❡✈❡r❡♠♦s p(x)₆=q(x)✳

❊①❡♠♣❧♦ ✶✳✶✳ ❖s ♣♦❧✐♥ô♠✐♦sp(x) = 5₋x2_{+ 4x−5x}3_{+ 6x}4_+x5 _{❡ q(x) =x}5_−5x3₊

5 + 4x₋x2_{+ 6x}4 _{sã♦ ✐❣✉❛✐s✱ ♣♦rq✉❡ ♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s a}

j ❞❛ j−és✐♠❛ ♣♦tê♥❝✐❛ xj✱ ❝♦♠ j _{∈ {}0,1,2,3,4,5_}✱ sã♦✿ a0 = 5✱ a1 = 4✱ a2 =−1✱a3 =−5✱ a4 = 6✱ a5 = 1✳

❙❡ ❡s❝r❡✈❡r♠♦s ♦s ♣♦❧✐♥ô♠✐♦s ❛❝✐♠❛ ❝♦♠ ❛s ♣♦tê♥❝✐❛s ❞❡ x ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡✱

✈✐s✉❛❧✐③❛r❡♠♦s ✐♠❡❞✐❛t❛♠❡♥t❡ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❡❧❡s✱ ♣♦✐s

p(x) =q(x) = 5 + 4x₋x2₋5x3+ 6x4+x5.

❊①❡♠♣❧♦ ✶✳✷✳ ❖s ♣♦❧✐♥ô♠✐♦sp(x) =₋x+4x2₋3x3+5x4❡q(x) = 4₋x+4x2₋3x3+5x4

sã♦ ❞✐❢❡r❡♥t❡s✱ ✈✐st♦ q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦s t❡r♠♦s ❝♦♥st❛♥t❡s ❞♦s ♣♦❧✐♥ô♠✐♦s p(x) ❡

q(x) sã♦ ❞✐❢❡r❡♥t❡s✱ a0 = 0 ❡b0 = 4✳

❊♠ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✱ p(x) _6≡ 0✱ ✷ ❛❧❣✉♠ ❝♦❡✜❝✐❡♥t❡ ❞❡✈❡rá

s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ♣♦rt❛♥t♦ ❤❛✈❡rá ✉♠ ♠❛✐♦r ♥ú♠❡r♦ ♥❛t✉r❛❧ n t❛❧ q✉❡an6= 0✳ ❉❡✲ ✜♥✐r❡♠♦s ♦ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦ ♣♦rn ❡✱ ♥❡st❡ ❝❛s♦✱ ans❡rá ❞❡♥♦♠✐♥❛❞♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ p(x)✳

❖s ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ n ❝♦♠ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r an = 1 s❡rã♦ ❞❡♥♦♠✐♥❛❞♦s ♣♦❧✐♥ô✲ ♠✐♦s ♠ô♥✐❝♦s✳

❖❜s❡r✈❛çã♦ ✶✳✶✳ ◆ã♦ s❡ ❞❡✜♥❡ ♦ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ✭0(x)_≡0✮✳ ✸

❯s❛r❡♠♦s ♦ sí♠❜♦❧♦ ❣r❛✉(p(x))♣❛r❛ ❞❡♥♦t❛r ♦ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦ p(x)✳

❊①❡♠♣❧♦ ✶✳✸✳ ❖ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡p(x) = 7♥ã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❡ ❣r❛✉(p(x)) = 0✳ ❖ ♣♦❧✐♥ô♠✐♦w(x) = 5₋x2+ 4x₋5x3+ 6x4+x5 t❡♠ ❣r❛✉ 5 ❡ é ♠ô♥✐❝♦✱ ❡♥q✉❛♥t♦

q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ v(x) = 4₋x+ 4x2₋3x3+ 5x4 t❡♠ ❣r❛✉ 4 ❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r a₄ = 5✳

▲❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ ❖❜s❡r✈❛çã♦ ✶✳✶✱ s❛❧✐❡♥t❛♠♦s q✉❡✿

❣r❛✉(p(x)) = 0 s❡✱ s❡ s♦♠❡♥t❡ s❡✱f(x) = a_{0 6}= 0✱ a_{0 ∈}R ✳

P♦❧✐♥ô♠✐♦s ❡ ♦♣❡r❛çõ❡s ✷✺

✶✳✶✳✶ ❆❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s

❉❡✜♥✐çã♦ ✶✳✸✳ ❉❡✜♥✐r❡♠♦s ❛ ❛❞✐çã♦ ❞♦s ♣♦❧✐♥ô♠✐♦s p(x) = n X

j=0

a_jxj ❡ q(x) = m

X

j=0

b_jxj ♣♦r

p(x) +q(x) = M X

j=0

(aj +bj)xj,

♦♥❞❡ M =✹max_{❣r❛✉(p(x)),❣r❛✉(q(x))_}✳

P❛r❛ ♦ ❛❧✉♥♦✱ é ✐♠♣♦rt❛♥t❡ ❧❡♠❜r❛r q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡ra, b_∈R✱ _{a−b=a+ (−b)}✳ ❱❡❥❛♠♦s ♦s s❡❣✉✐♥t❡s ❡①❡♠♣❧♦s✳

❊①❡♠♣❧♦ ✶✳✹✳ ❙❡❥❛♠ p(x) = 4x4 ₋ 3x2 + 7x+ 1✱ q(x) = 5x4 ₋6x₋ 1 ❡ w(x) =

−3x4+ 6x3 + 2x2+ 3✳ ❊♥tã♦✱

p(x) +q(x) = (4 + 5)x4+ (₋3 + 0)x2+ (7₋6)x+ (1₋1) = 9x4₋3x2+x,

p(x)+w(x) = (4₋3)x4+(0+6)x3+(₋3+2)x2+(7+0)x+(1+3) =x4+6x3₋x2+7x+4,

q(x)+w(x) = (5₋3)x4+(0+6)x3+(0+2)x2+(₋6+0)x+(₋1+3) = 2x4+6x3+2x2₋6x+2.

❊①❡♠♣❧♦ ✶✳✺✳ ❙❡❥❛♠ p(x) = 4x4 ₋ 3x2 + 7x+ 1✱ q(x) = 5x2 ₋6x₋ 1 ❡ w(x) = 4x5+ 6x3+ 2x2+ 3✳ ❊♥tã♦✱

p(x) +q(x) = (4 + 0)x4+ (₋3 + 5)x2+ (7₋6)x+ (1₋1) = 4x4+ 2x2+x,

p(x)+w(x) = (0+4)x5+(4+0)x4+(0+6)x3+(₋3+2)x2+(7+0)x+(1+3) = 4x5+4x4+6x3₋x2+7x+4,

q(x)+w(x) = (0+4)x5+(0+6)x3+(5+2)x2+(₋6+0)x+(₋1+3) = 4x5+6x3+7x2₋6x+2.

◆♦ ❊①❡♠♣❧♦ ✶✳✹✱ s♦♠❛♠♦s ♣♦❧✐♥ô♠✐♦s q✉❡ ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❣r❛✉ ✭❣r❛✉(p(x)) =

❣r❛✉(q(x)) = ❣r❛✉(w(x)) = 4✮✱ ❡♥q✉❛♥t♦ q✉❡✱ ♥♦ ❊①❡♠♣❧♦ ✶✳✺✱ s♦♠❛♠♦s ♣♦❧✐♥ô♠✐♦s

q✉❡ ♣♦ss✉❡♠ ❣r❛✉s ❞✐❢❡r❡♥t❡s ✭❣r❛✉(p(x)) = 4✱ ❣r❛✉(q(x)) = 2✱ ❣r❛✉(w(x)) = 5✮✳

◆❛ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❣r❛✉✿ Pr♦♣♦s✐çã♦ ✶✳✶✳ ❙❡❥❛♠ p(x) =

n X

j=0

ajxj✱ ❝♦♠ an6= 0✱ ❡ q(x) = m X

j=0

b_jxj✱ ❝♦♠ bm 6= 0✳ ❙❡ p(x) +q(x)₆= 0✱ ❡♥tã♦

❣r❛✉(p(x) +q(x))_≤max_{❣r❛✉(p(x)),❣r❛✉(q(x))_}= max_{n, m_}.

❆ ✐❣✉❛❧❞❛❞❡ s❡rá ✈á❧✐❞❛ s❡♠♣r❡ q✉❡ ❣r❛✉(p(x))₆=❣r❛✉(q(x))✳

✷✻ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s

◆♦ ❊①❡♠♣❧♦ ✶✳✺✱ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶ é ❢❛❝✐❧♠❡♥t❡ ❝♦♥st❛t❛❞❛✳

❆ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s t❡♠ ❞✐✈❡rs❛s ♣r♦♣r✐❡❞❛❞❡s✱ q✉❡ sã♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❞❛ ❛❞✐çã♦ ♥♦ ❝♦♥❥✉♥t♦ R✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ❛ s❡❣✉✐r✳

Pr♦♣r✐❡❞❛❞❡s ❞❛ ❛❞✐çã♦

❈♦♥s✐❞❡r❡♠♦s ♦s ♣♦❧✐♥ô♠✐♦s p(x) = n X

j=0

ajxj✱ q(x) = m X

j=0

b_jxj ❡w(x) = l X

j=0

cjxj✳ ❆✶✳ ❈♦♠✉t❛t✐✈❛✿

p(x) +q(x) = q(x) +p(x),

♣♦✐s✱ ♣❛r❛ q✉❛✐sq✉❡raj, bj ∈R✱ ❝♦♠0≤j ≤max{n, m}✱ t❡♠♦saj+bj =bj+aj✳ ❆✷✳ ❆ss♦❝✐❛t✐✈❛✿

(p(x) +q(x)) +w(x) = p(x) + (q(x) +w(x)),

♣♦✐s✱ ♣❛r❛ q✉❛✐sq✉❡r aj, bj, cj ∈ R✱ ❝♦♠ 0≤ j ≤ max{n, m, l}✱ t❡♠♦s (aj +bj) +

cj =aj+ (bj +cj)✳

❆✸✳ ❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✿ ❖ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ 0 =Pn

j=00xj s❛t✐s❢❛③ p(x) + 0 = 0 +p(x)♣♦✐s✱

♣❛r❛ 0_≤j _≤n ❡ aj ∈R✱ t❡♠♦s aj = 0 +aj✳ ❆✹✳ ❊①✐stê♥❝✐❛ ❞❡ s✐♠étr✐❝♦

❉❛❞♦ p(x) = Pn

j=0ajxj✱ ♦ ♣♦❧✐♥ô♠✐♦ −p(x) = Pn_j=0(−aj)xj é ♦ s✐♠étr✐❝♦ ❞❡

p(x)✱ s❡♥❞♦

p(x) + (₋p(x)) = n X

j=0

0xj,

♣♦✐s (a_j) + (₋a_j) = 0✱ ♣❛r❛ q✉❛❧q✉❡r a_{j ∈}R✱ _0≤j _≤n✳

❊①❡♠♣❧♦ ✶✳✻✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ♣♦❧✐♥ô♠✐♦sp(x) = 4x4₋3x2+7x+1✱q(x) = 5x2₋6x₋1

❡ w(x) = 4x5+ 6x3+ 2x2+ 3 ❞♦ ❊①❡♠♣❧♦ ✶✳✺✳

◆♦ ❊①❡♠♣❧♦ ✶✳✺✱ ❞❡t❡r♠✐♥❛♠♦s p(x) +q(x) = 4x4+ 2x2+x✳ ❆ss✐♠✱

(p(x) +q(x)) +w(x) = (4x4+ 2x2+x) + (4x5 + 6x3+ 2x2 + 3)

= (0 + 4)x5 + (4 + 0)x4+ (0 + 6)x3+ (2 + 2)x2+ (1 + 0)x+ 3

= 4x5+ 4x4+ 6x3+ 4x2+x+ 3.

❉❡t❡r♠✐♥❛♠♦s✱ t❛♠❜é♠✱ q(x) +w(x) = 4x5+ 6x3 + 7x2₋6x+ 2✳ ❆ss✐♠✱

p(x) + (q(x) +w(x)) = (4x4₋3x2+ 7x+ 1) + (4x5+ 6x3+ 7x2₋6x+ 2)

P♦❧✐♥ô♠✐♦s ❡ ♦♣❡r❛çõ❡s ✷✼

❖✉ s❡❥❛✱ (p(x) +q(x)) +w(x) =p(x) + (q(x) +w(x))✱ ❝♦♠♦ ♥♦s ❞✐③ ❛ ♣r♦♣r✐❡❞❛❞❡

❛ss♦❝✐❛t✐✈❛✳ ❆s ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s t❛♠❜é♠ sã♦ ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛❞❛s ❝♦♥s✐❞❡r❛♥❞♦ ♦s ♣♦❧✐♥ô♠✐♦s ❛❝✐♠❛✳

✶✳✶✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ P♦❧✐♥ô♠✐♦s

❉❡✜♥✐çã♦ ✶✳✹✳ ❉❡✜♥✐r❡♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦s ♣♦❧✐♥ô♠✐♦sp(x) = n X

j=0

ajxj ❡ q(x) = m

X

j=0

bjxj ♣♦r

p(x).q(x) = n+m X

j=0

cjxj,

s❡♥❞♦

c₀ = a₀.b₀

c1 = a0.b1+a1.b0

c₂ = a₀.b₂+a₁.b₁+a₂.b₀

✳✳✳

cj = a0.bj+a1.bj−1+· · ·+aj.b0 =

X

λ+µ=j

aλ.bµ

✳✳✳

cn+m = an.bm.

◆❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❣r❛✉✿ Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡❥❛♠ p(x) =

n X

j=0

a_jxj✱ ❝♦♠ an6= 0✱ ❡ q(x) = m X

j=0

b_jxj✱ ❝♦♠ bm 6= 0✳ ❊♥tã♦✱

❣r❛✉(p(x).q(x)) =n+m,

♣♦✐s ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ p(x).q(x) é cn+m =an.bm 6= 0✳

❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

Pr♦♣r✐❡❞❛❞❡s ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦ ❙❡❥❛♠ p(x) =

n X

j=0

ajxj✱ q(x) = m X

j=0

bjxj ❡ w(x) = l X

j=0

✷✽ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s

▼✶✳ ❈♦♠✉t❛t✐✈❛✿

p(x).q(x) = q(x).p(x),

♣♦✐s✱ ♣❛r❛ t♦❞♦ j _{∈ {}0,1,_{· · ·} , n+m_}✱ ✈❛❧❡ ❛ ✐❞❡♥t✐❞❛❞❡✿

X

λ+µ=j

aλbµ = X

λ+µ=j

bµ.aλ.

▼✷✳ ❆ss♦❝✐❛t✐✈❛✿

(p(x).q(x)).w(x) = p(x).(q(x).w(x)).

❈❛❜❡ ♦❜s❡r✈❛r q✉❡✱ ❡♠ ✈✐rt✉❞❡ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ t❡♠♦s✿

• P❛r❛ q✉❛✐sq✉❡r j, k _∈N✱ ✈❛❧❡ ❛ ✐❞❡♥t✐❞❛❞❡✿ _xj_.xk _=xj+k✳

• ❙❡p(x) =a₀ ❡q(x) = b₀+b₁.x+_{· · ·}+bm.xm✱ ❡♥tã♦

p(x).q(x) = a0.q(x) = a0.

m X

j=0

bjxj = m X

k=0

a0bkxk

= (a0.b0) + (a0.b1)x+· · ·+ (a0.bm)xm, ♣♦✐s✱ ♥❡st❡ ❝❛s♦✱n = 0 ❡ cj =a0bj✱ ♣❛r❛ t♦❞♦ j ∈N✳

❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♥s✐❞❡r❛♥❞♦ p(x) = 1✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s t❡♠ ❛

s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿

▼✸✳ ❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✿ ✶✳ q✭①✮ ❂ q✭①✮✱ ♣❛r❛ q✉❛❧q✉❡r ♣♦❧✐♥ô♠✐♦q(x)✳

❖ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ é t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞♦ ✉♥✐❞❛❞❡✳

❈♦♠❜✐♥❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❝♦♠ ♦ ❢❛t♦ ❞❛ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦rr❡s♣♦♥❞❡r ❛ ❛❞✐❝✐♦♥❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ♣♦tê♥❝✐❛s ❞❡ x ❞❡ ♠❡s♠♦ ❡①♣♦❡♥t❡ ❡♠

❛♠❜♦s ♦s ♣♦❧✐♥ô♠✐♦s✱ ♦❜t❡♠♦s ♠❛✐s ✉♠❛ ♣r♦♣r✐❡❞❛❞❡✱ ❛ q✉❛❧ ❡♥✈♦❧✈❡ ❛s ❞✉❛s ♦♣❡r❛çõ❡s✳ Pr♦♣r✐❡❞❛❞❡ ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦

❙❡❥❛♠ p(x) = n X

j=0

ajxj✱ q(x) = m X

j=0

bjxj ❡ w(x) = l X

j=0

cjxj ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐✲ ❡♥t❡s ❡♠ R✳

❆▼✳ ❉✐str✐❜✉t✐✈❛✿

(p(x) +q(x)).w(x) = p(x).w(x) +q(x).w(x).

P❛r❛ ♦ ❛❧✉♥♦✱ é ✐♠♣♦rt❛♥t❡ ❧❡♠❜r❛r q✉❡ ❛ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ R tê♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛✱ ♦✉ s❡❥❛✱ (a+b).c=a.c+b.c✱ ♣❛r❛ q✉❛✐sq✉❡r a, b, c_∈R✳

❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✷✾

❊①❡♠♣❧♦ ✶✳✼✳ ❙❡❥❛♠ p(x) = 4x3 ₋3x2 + 4x+ 5 ❡ q(x) = 2x2 ₋ 5x ₋2 ✳ ❊♥tã♦✱

✉t✐❧✐③❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❞✉❛s ♦♣❡r❛çõ❡s✱ ♦❜t❡♠♦s✿

p(x).q(x) = (4x3₋3x2+ 4x+ 5).(2x2₋5x₋2)

1

= 4x3.(2x2₋5x₋2) + (₋3x2).(2x2₋5x₋2) + 4x.(2x2₋5x₋2) + 5.(2x2₋5x₋2)

2

= (8x5₋20x4₋8x3) + (₋6x4+ 15x3 + 6x2) + (8x3₋20x2₋8x) + (10x2₋25x₋10)

3

= 8x5+ (₋20₋6)x4+ (₋8 + 15 + 8)x3+ (6₋20 + 10)x2+ (₋8₋25)x₋10

4

= 8x5₋26x6+ 15x3₋4x2₋33x₋10.

❖❜s❡r✈❛♠♦s q✉❡ ❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ ❢♦r❛♠ ♦❜t✐❞❛s ❞❛s s❡❣✉✐♥t❡s ❢♦r♠❛s✿ ✶✳ ❉✐str✐❜✉✐♥❞♦ ❛s ♣❛r❝❡❧❛s ❞❡ p(x) ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r q(x)❀

✷✳ ❉✐str✐❜✉✐♥❞♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❝❛❞❛ t❡r♠♦ ❞❡p(x) ♣♦rq(x)❀

✸✳ ❯t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s❀

✹✳ ❋❛③❡♥❞♦ ❛ ❛❞✐çã♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ♣♦tê♥❝✐❛s ❞❡x ❞❡ ♠❡s♠♦ ❡①♣♦❡♥t❡✳

✶✳✷ ❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s

◆❛s ♣ró①✐♠❛s s❡çõ❡s✱ ❛♣r❡♥❞❡r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ♦ ❛❧❣♦r✐t♠♦ ❡✉✲ ❝❧✐❞✐❛♥♦ ♣❛r❛ ♣♦❧✐♥ô♠✐♦s✳ ❱❡r❡♠♦s✱ t❛♠❜é♠✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ r❛✐③ r❡❛❧ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ❡ r❡❧❛❝✐♦♥❛r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ r❛✐③ r❡❛❧ α ❝♦♠ ❛ ❞✐✈✐s✐❜✐✲

❧✐❞❛❞❡ ♣♦r x₋α✳ ▼❛✐s ❛✐♥❞❛✱ r❡❧❛❝✐♦♥❛r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ n r❛í③❡s r❡❛✐s ❞✐st✐♥t❛s α1, α2,· · ·, αn✱ q✉❛♥❞♦ ♦ ♣♦❧✐♥ô♠✐♦ t✐✈❡r ❣r❛✉ ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛n✱ ❝♦♠ ❛ ❞✐✈✐s✐✲ ❜✐❧✐❞❛❞❡ ♣♦r (x₋α1)· · ·(x−αn)✳ P♦r ✜♠✱ ♠♦str❛r❡♠♦s ❝♦♠♦ ❞❡t❡r♠✐♥❛r ❛s ♣♦ssí✈❡✐s r❛í③❡s r❛❝✐♦♥❛✐s ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✐♥t❡✐r♦s✳

◆♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ ❝♦♥❝❡✐t♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✶✳✺✳ ❙❡❥❛♠ p(x)❡ q(x) ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ R✱ ❝♦♠ _{q(x) 6= 0}✳ ❉✐r❡♠♦s q✉❡ q(x) ❞✐✈✐❞❡ p(x) s❡ ❡①✐st✐r ✉♠ ♣♦❧✐♥ô♠✐♦ h(x) t❛❧ q✉❡

p(x) =q(x).h(x).

❉✐r❡♠♦s t❛♠❜é♠ q✉❡p(x) é ♠ú❧t✐♣❧♦ ❞❡ q(x) ♦✉ q✉❡ p(x) é ❞✐✈✐sí✈❡❧ ♣♦r q(x)✳

❊①❡♠♣❧♦ ✶✳✽✳ ✶✳ ❈♦♠♦ x2 _{−16 = (x−4)(x+ 4)✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✶✳✺✱ x−4 ❞✐✈✐❞❡}

x2 _{−16✳ ◆❡st❡ ❝❛s♦✱ h(x) = x+ 4✳ ◆♦t❡ q✉❡✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛✱ x+ 4 ❞✐✈✐❞❡}

x2_−16✳

✷✳ ❖ ♣♦❧✐♥ô♠✐♦x4+ 5x5+ 6 ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

x4+ 5x2+ 6 = (x2+ 3)(x2+ 2).

✸✵ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s

✸✳ ❉❛❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s m _≤ n✱ ♦ ♣♦❧✐♥ô♠✐♦ xm ❞✐✈✐❞❡ xn ♣♦✐s✱ t♦♠❛♥❞♦ r =

n₋m _≥0✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

xn =xm+r=xm.xr.

◆❛ s✉❜s❡çã♦ ♣r❡❝❡❞❡♥t❡✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ♥❛ Pr♦♣♦s✐çã♦ ✶✳✷✱ ✈✐♠♦s q✉❡ s❡ p(x)❡

q(x) ❢♦r❡♠ ♣♦❧✐♥ô♠✐♦s ♥ã♦ ♥✉❧♦s✱ ❡♥tã♦

❣r❛✉(p(x).q(x)) =❣r❛✉(p(x)) +❣r❛✉(q(x)).

❊♠ ✈✐rt✉❞❡ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

Pr♦♣♦s✐çã♦ ✶✳✸✳ ❙❡ p(x) ❡q(x)❢♦r❡♠ ♣♦❧✐♥ô♠✐♦s ♥ã♦ ♥✉❧♦s ❡q(x) ❞✐✈✐❞✐rp(x)✱ ❡♥tã♦

❣r❛✉(p(x))_≥❣r❛✉(q(x))✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ q(x) ❞✐✈✐❞❡ p(x) ❡ ❛♠❜♦s sã♦ ♥ã♦ ♥✉❧♦s✱ ❡①✐st❡

✉♠ ♣♦❧✐♥ô♠✐♦ h(x) ♥ã♦ ♥✉❧♦ t❛❧ q✉❡ p(x) = q(x).h(x)✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❣r❛✉

s✉♣r❛❝✐t❛❞❛✱ t❡♠♦s

❣r❛✉(p(x)) = ❣r❛✉(q(x).h(x)) = ❣r❛✉q(x) +❣r❛✉h(x)_≥❣r❛✉q(x),

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

◗✉❡ ✜q✉❡ ❝❧❛r♦ q✉❡ ♥❡♠ s❡♠♣r❡ ✉♠ ♣♦❧✐♥ô♠✐♦ é ♠ú❧t✐♣❧♦ ❞❡ ✉♠ ♦✉tr♦ ♣♦❧✐♥ô♠✐♦ q✉❛❧q✉❡r ❞❡ ❣r❛✉ ✐♥❢❡r✐♦r✳ ❱❡r❡♠♦s✱ ❛ s❡❣✉✐r✱ ✉♠ ❡①❡♠♣❧♦ ♦♥❞❡ ❡ss❛ ❛✜r♠❛çã♦ s❡rá ❝♦♥st❛t❛❞❛✳ ■♥❢♦r♠❛♠♦s✱ ❞❡ ❛♥t❡♠ã♦✱ q✉❡ ❛ ❡str❛té❣✐❛ ✉s❛❞❛ ♣❛r❛ r❡s♣♦♥❞❡r ❛ ♣❡r✲ ❣✉♥t❛ ❞♦ ❡①❡♠♣❧♦ s❡❣✉✐♥t❡ é ❛ ❘❡❞✉çã♦ ❛♦ ❆❜s✉r❞♦✳ ❊♠❜♦r❛ ❡ss❛ ❡str❛té❣✐❛ ♥ã♦ s❡❥❛ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❝♦♥t❛❞❛ ❛♦s ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ s❡❝✉♥❞ár✐♦✱ ♥ã♦ ✐❞❡♥t✐✜❝❛♠♦s ♣r♦❜❧❡♠❛s ❛♦ ❛♣r❡s❡♥tá✲❧❛ ❛♦s ♠❡s♠♦s✳ ❆❧é♠ ❞✐ss♦✱ ❥✉❧❣❛♠♦s q✉❡ ❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ❘❡❞✉çã♦ ❛♦ ❆❜s✉r❞♦ s❡❥❛ ❜❡♠ ♣❡rt✐♥❡♥t❡ ❛♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡♠ t♦❞❛s ❛s ❡s❝♦❧❛s✳ ❊①❡♠♣❧♦ ✶✳✾✳ ❖ ♣♦❧✐♥ô♠✐♦ q(x) =x+ 4 ❞✐✈✐❞❡ ♦ ♣♦❧✐♥ô♠✐♦ p(x) =x2+ 3x+ 2❄ ❖✉

s❡❥❛✱ ❤á ❛❧❣✉♠ ♣♦❧✐♥ô♠✐♦ h(x) t❛❧ q✉❡ x2+ 3x+ 2 = (x+ 4)h(x)❄

❖r❛✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ♦ t❛❧ ♣♦❧✐♥ô♠✐♦ h(x)✳ ❈♦♠♦ ❣r❛✉(x2 + 3x+ 2) = 2 ❡

❣r❛✉(x+ 4) = 1✱ ❡st❡ ♣♦❧✐♥ô♠✐♦ h(x) ❞❡✈❡ t❡r ❣r❛✉ ✐❣✉❛❧ ❛ 1✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❞✐③❡r

q✉❡

h(x) = ax+b, a₆= 0 ❡ a, b_∈R_. ❆ss✐♠✱

x2+ 3x+ 2 = (x+ 4)(ax+b) =ax2+ 4ax+bx+ 4b=ax2+ (4a+b)x+ 4b.

❉❡✈❡♠♦s✱ ♣♦✐s✱ t❡r✿

a = 1, 4a+b= 3, 4b = 2

b = 1 2

.

❈♦♠♦ a= 1 ❡b = 1₂✱ ❛ ❡q✉❛çã♦ 4a+b= 3 ♥♦s ❞✐③ q✉❡ 3 = 4a+b= 4.1 + 1₂ = 9₂✱ ♦

q✉❡ é ❛❜s✉r❞♦✳

❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✸✶

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♥♦s ❛♣r❡s❡♥t❛ ♦ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✳ P❛rt❡ ❞❛ ❞❡♠♦♥s✲ tr❛çã♦ ❞♦ ♠❡s♠♦ s❡rá ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ ♦ ❣r❛✉ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦❧✐♥ô♠✐♦✳ ❙❡♥❞♦ ❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ✉♠ ♠ét♦❞♦ r❡str✐t♦ ❛♦ ❡♥s✐♥♦ s✉♣❡r✐♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ✐♥❢♦r♠❛♠♦s q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦ ❛♣❡♥❛s ♣❛r❛ ❛✉①✐❧✐❛r ♦ ❧❡✐t♦r ✲ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ❥á ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ t❛❧ ♠ét♦❞♦✱ ♥❛ r❡✈✐sã♦ ❞❛ t❡♦r✐❛ ❞❡ ♣♦❧✐♥ô♠✐♦s✳ ■♥❞✐❝❛♠♦s ❛ r❡❢❡rê♥❝✐❛ ❬✽❪✱ ♣❛r❛ ♦ ❧❡✐t♦r✲❛❧✉♥♦ ✐♥t❡r❡ss❛❞♦ ❡♠ ❡♥t❡♥❞❡r ♥♦ q✉❡ ❝♦♥s✐st❡ ✉♠❛ ♣r♦✈❛ ♣♦r ✐♥❞✉çã♦✳

❚❡♦r❡♠❛ ✶✳✶ ✭❉✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✮✳ ❙❡❥❛♠ p(x)❡ q(x) ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠

R✱ ❝♦♠ _{q(x) 6= 0}✳ ❊♥tã♦✱ ❡①✐st❡♠ ♣♦❧✐♥ô♠✐♦s _h(x) ❡ _r(x)✱ ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱ t❛✐s q✉❡

p(x) = h(x)q(x) +r(x), ✭✶✳✶✮

♦♥❞❡ r(x) = 0 ♦✉ ❣r❛✉(r(x))<❣r❛✉(q(x))✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ q(x) ₆= 0✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ q(x) = b₀ +b₁x+ _{· · ·} +

bm−1xm−1+bmxm✱ ♦♥❞❡ m=❣r❛✉q(x)✳

❉❡✈❡♠♦s✱ ❡♥tã♦✱ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞♦s ♣♦❧✐♥ô♠✐♦sh(x)❡r(x)♣❛r❛

♦s q✉❛✐s t❡♥❤❛♠♦s ✭✶✳✶✮✳ P♦✐s ❜❡♠✱ ❝♦♠❡ç❛r❡♠♦s ♠♦str❛♥❞♦ ❛ ❡①✐stê♥❝✐❛✳ ✭❊①✐stê♥❝✐❛✮ ❙❡p(x) = 0✱ ❜❛st❛ t♦♠❛r h(x) = r(x) = 0✳

❙✉♣♦♥❤❛♠♦s q✉❡p(x)₆= 0✳ ❙❡❥❛♠n =❣r❛✉p(x)❡p(x) =a₀+a₁x+_{· · ·}+an−1xn−1+

anxn✱ ❝♦♠ an 6= 0✳

•❙❡ n < m✱ t♦♠❡ h(x) = 0 ❡ r(x) =p(x)✳

•❙✉♣♦♥❤❛♠♦s q✉❡ n _≥m✳ ◆❡st❡ ❝❛s♦✱ ❛ ✜♠ ❞❡ ❝♦♥❝❧✉✐r♠♦s ♦ ❞❡s❡❥❛❞♦✱ ❛r❣✉♠❡♥✲

t❛r❡♠♦s ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n=❣r❛✉p(x)✳

❈♦♠ ❡❢❡✐t♦✱ s❡n = 0✱ ❡♥tã♦ 0 = n _≥m =❣r❛✉q(x) ❡✱ ♣♦rt❛♥t♦✱ m = 0✱ p(x) = a₀

❡ q(x) = b₀✳ ❆ss✐♠✱ p(x) =a₀b−1

0 q(x)✱ ❝♦♠ h(x) =a0b−01 ❡r(x) = 0✳

❙✉♣♦♥❤❛♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈á❧✐❞♦ ♣❛r❛ ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ ♠❡♥♦r ❞♦ q✉❡

n =❣r❛✉(p(x))❡ ✈❛♠♦s ♠♦str❛r q✉❡ ✈❛❧❡ ♣❛r❛ p(x)✱ q✉❡ t❡♠ ❣r❛✉ n✳

❉❡✜♥❛♠♦s

p₁(x) = p(x)₋anb−m1x

n−m_{q(x). ✭✶✳✷✮}

❖❜s❡r✈❡ q✉❡ ❣r❛✉(p1(x)) < ❣r❛✉(p(x))✱ ✉♠❛ ✈❡③ q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ anb−m1xn−mq(x) t❡♠ ❣r❛✉ n ❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r an✳ P♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ ❡①✐st❡♠ ♣♦❧✐♥ô♠✐♦s h1(x) ❡

r1(x) t❛✐s q✉❡

p1(x) = h1(x).q(x) +r1(x), ✭✶✳✸✮

❝♦♠ r1(x) = 0 ♦✉ ❣r❛✉(r1(x))<❣r❛✉(q(x))✳

P♦r ✭✶✳✷✮ ❡ ✭✶✳✸✮✱ t❡♠♦s

p(x) = p1(x) +anb−m1xn−mq(x) = (h1(x)q(x) +r1(x)) +anb−m1x

n−m_q(x)

✸✷ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s

❊♥tã♦✱ ❜❛st❛ t♦♠❛r h(x) = h1(x) +anbm−1xn−m ❡r(x) = r1(x)✳

❆❣♦r❛✱ ♠♦str❛r❡♠♦s ❛ ✉♥✐❝✐❞❛❞❡✳

✭❯♥✐❝✐❞❛❞❡✮ ❙❡❥❛♠ h1(x), r1(x), h2(x), r2(x) ♣♦❧✐♥ô♠✐♦s t❛✐s q✉❡

p(x) =h₁(x)q(x) +r₁(x)(=⋆)h₂(x).q(x) +r₂(x), ✭✶✳✹✮

♦♥❞❡ ₍

r₁(x) = 0 ♦✉ ❣r❛✉(r₁(x))<❣r❛✉(q(x)) ❡

r₂(x) = 0 ♦✉ ❣r❛✉(r₂(x))<❣r❛✉(q(x)). ✭⋆ ⋆✮

❉❡ (⋆)✱ s❡❣✉❡ q✉❡ (h₁(x)₋h₂(x))q(x) =r₂(x)₋r₁(x)✳

❙❡ h₁(x) ₆= h₂(x)✱ ❡♥tã♦ h₁(x) ₋ h₂(x) ₆= 0, ❧♦❣♦ r₂(x) ₋ r₁(x) ₆= 0 ❡✱ ♣❡❧❛

Pr♦♣♦s✐çã♦✶✳✷✱ ❝♦♥❝❧✉í♠♦s q✉❡

❣r❛✉(q(x))_≤❣r❛✉(r2(x)−r1(x)) (⋆⋆)

< ❣r❛✉(q(x)).

❊✐s✱ ♣♦✐s✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❊♥tã♦✱h₁(x) = h₂(x)❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ r₂(x) =r₁(x)✳

❉❡✜♥✐çã♦ ✶✳✻✳ ❙❡❥❛♠p(x)✱ q(x)✱h(x)❡ r(x)❝♦♠♦ ♥♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳ ❈❤❛♠❛r❡♠♦s

p(x) ❞❡ ❞✐✈✐❞❡♥❞♦✱q(x) ❞❡ ❞✐✈✐s♦r✱ h(x) ❞❡ q✉♦❝✐❡♥t❡ ❡ r(x) ❞❡ r❡st♦✳

❉❡✈❡♠♦s ♣r❡st❛r ❛t❡♥çã♦ ❛♦s ❣r❛✉s ❞♦ ❞✐✈✐❞❡♥❞♦✱ ❞♦ ❞✐✈✐s♦r ❡ ❞♦ r❡st♦ ♣❛r❛ ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦✳

❱❡r❡♠♦s ❝♦♠♦ ❞❡t❡r♠✐♥❛r ♦ q✉♦❝✐❡♥t❡ h(x) ❡ ♦ r❡st♦ r(x) ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞♦

♣♦❧✐♥ô♠✐♦p(x)♣♦rq(x)₆= 0✳ ❊❧❛❜♦r❛r❡♠♦s ✉♠❛ t❛❜❡❧❛✱ ✐❧✉str❛♥❞♦ ♦s ❝á❧❝✉❧♦s ♣❛ss♦ ❛

♣❛ss♦✳ ❖s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r ❝♦♥s✐st❡♠ ❞❡ ❛r♠❛r ❡ ❡❢❡t✉❛r✱ ❝♦♥❢♦r♠❡ ♦ ♠♦❞❡❧♦✿

p(x) q(x)

✳✳✳ h(x)

r(x)

❊①❡♠♣❧♦ ✶✳✶✵✳ ❙❡❥❛♠p(x) = 5x+2❡q(x) =x3+2x+1✳ ❈♦♠♦ ❣r❛✉(p(x)) = 1<3 =

❣r❛✉(q(x))✱ ♥❛❞❛ t❡♠♦s ❛ ❢❛③❡r✳ ❖ q✉♦❝✐❡♥t❡ éh(x) = 0❡ ♦ r❡st♦ ér(x) = p(x) = 5x+2✳ 5x+ 2 x3+ 2x+ 1

−0 0 5x+ 2

❊①❡♠♣❧♦ ✶✳✶✶✳ ❙❡❥❛♠ p(x) = 2x2 + 4x+ 3 ❡ q(x) = x2 + 3x+ 1✳

✶✳ ❖ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ p(x) é 2x2 ❡ ♦ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ q(x) é

x2✳ ❖ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ ❞❡ 2x2 ♣♦r x2 é h₁(x) = 2✳

✷✳ ❋❛③❡♥❞♦ ♦ ❝á❧❝✉❧♦✱ ♦❜t❡♠♦s✿

r₁(x) =p(x)₋h₁(x)q(x) = (2x2+ 4x+ 3)₋2x2₋6x₋2 =₋2x+ 1 2x2+ 4x+ 3 x2+ 3x+ 1

−2x2₋6x₋2 2

❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✸✸

✸✳ ❈♦♠♦ ❣r❛✉(r1(x)) = 1 < 2 = ❣r❛✉(q(x))✱ ♥ã♦ ♣♦❞❡♠♦s ❝♦♥t✐♥✉❛r ❛ ❞✐✈✐sã♦ ❡✱

♣♦rt❛♥t♦✱ ♣❛r❛♠♦s ♦s ❝á❧❝✉❧♦s✳

✹✳ ❖❜t❡♠♦s✱ ♣♦✐s✱ h(x) =h1(x) = 2 ❡ r(x) = r1(x) = −2x+ 1✳

❊①❡♠♣❧♦ ✶✳✶✷✳ ❙❡❥❛♠ p(x) = 3x4+ 5x3+x2+ 2x₋3 ❡ q(x) =x2+ 3x+ 1✳

✶✳ ❖ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ p(x) é 3x4 ❡ ♦ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ q(x) é

x2✳ ❖ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ ❞❡ 3x4 ♣♦rx2 é h1(x) = 3x2✳

✷✳ ❋❛③❡♥❞♦ ♦ ❝á❧❝✉❧♦✱ ♦❜t❡♠♦s✿

r₁(x) =p(x)₋h₁(x)q(x) = (3x4+5x3+x2+2x₋3)₋3x4₋9x3₋3x2 =₋4x3₋2x2+2x₋3.

3x4+ 5x3+x2+ 2x₋3 x2+ 3x+ 1

−3x4₋9x3₋3x2 3x2

−4x3₋2x2+ 2x₋3

✸✳ ❈♦♠♦ ❣r❛✉(r₁(x)) = 3>2 = ❣r❛✉(q(x))✱ ❞❡✈❡♠♦s ❝♦♥t✐♥✉❛r ❛ ❞✐✈✐sã♦✱ ❞✐✈✐❞✐♥❞♦

r₁(x) ♣♦rq(x)✱ ♣♦✐s r₁(x)♥ã♦ é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✳

✹✳ ❖ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ r₁(x) é ₋4x3 ❡ ♦ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ q(x)

é x2✳ ❖ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ ❞❡₋4x3 ♣♦r x2 éh₂(x) = ₋4x✳

✺✳ ❋❛③❡♥❞♦ ♦ ❝á❧❝✉❧♦✱ ♦❜t❡♠♦s✿

r2(x) =r1(x)−h2(x)q(x) = (−4x3−2x2+2x−3)+4x3+12x2+4x= 10x2+6x−3.

3x4+ 5x3+x2+ 2x₋3 x2+ 3x+ 1

−3x4₋9x3₋3x2 3x2₋4x

−4x3₋2x2+ 2x₋3 4x3+ 12x2+ 4x

10x2+ 6x₋3

✻✳ ❈♦♠♦ ❣r❛✉(r2(x)) = 2 = ❣r❛✉(q(x))✱ ❞❡✈❡♠♦s ❝♦♥t✐♥✉❛r ❛ ❞✐✈✐sã♦✱ ❞✐✈✐❞✐♥❞♦

r2(x) ♣♦rq(x)✱ ♣♦✐s r2(x)♥ã♦ é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✳

✼✳ ❖ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡r2(x) é10x2 ❡ ♦ ♠♦♥ô♠✐♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ q(x) é

x2✳ ❖ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ ❞❡ 10x2 ♣♦r x2 é h3(x) = 10✳

✽✳ ❋❛③❡♥❞♦ ♦ ❝á❧❝✉❧♦✱ ♦❜t❡♠♦s✿

r₃(x) =r₂(x)₋h₃(x)q(x) = (10x2+ 6x₋3)₋10x2₋30x₋10 =₋24x₋13.

3x4+ 5x3+x2+ 2x₋3 x2+ 3x+ 1

−3x4₋9x3₋3x2 3x2₋4x+ 10

−4x3₋2x2+ 2x₋3 4x3+ 12x2 + 4x

10x2+ 6x₋3

−10x2₋30x₋10

✸✹ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s

✾✳ ❈♦♠♦ ❣r❛✉(r3(x)) = 1 < 2 = ❣r❛✉(q(x))✱ ♥ã♦ ♣♦❞❡♠♦s ❝♦♥t✐♥✉❛r ❛ ❞✐✈✐sã♦ ❡✱

♣♦rt❛♥t♦✱ ♣❛r❛♠♦s ♦s ❝á❧❝✉❧♦s✳

✶✵✳ ❖❜t❡♠♦s✱ ♣♦✐s✱ h(x) =h1(x) +h2(x) +h3(x) = 3x2 −4x+ 10 ❡ r(x) = r3(x) =

−24x₋13✳

✶✳✸ ❘❛✐③ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦

❙❡❥❛ α ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❆ ❛✈❛❧✐❛çã♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ p(x) = a0 +a1x+· · ·+

an−1xn−1 +anxn ❡♠ α é ❞❡✜♥✐❞❛ ♣♦r

p(α) =a₀+a₁α+_{· · ·}+an−1αn−1+anαn ∈R, ♦ q✉❡ ❡q✉✐✈❛❧❡ ❛ s✉❜st✐t✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧ x ❞♦ ♣♦❧✐♥ô♠✐♦p(x) ♣♦rα✳

❉❡✜♥✐çã♦ ✶✳✼✳ ❙❡❥❛ p(x) = a0 +a1x +· · · +an−1xn−1 + anxn ✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ R t❛❧ q✉❡ ❣r❛✉_(p(x))≥1✳ ❉✐r❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ _α é ✉♠❛ r❛✐③ ❞❡ p(x) q✉❛♥❞♦ p(α) = 0✳

❊①❡♠♣❧♦ ✶✳✶✸✳ ✶ é r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦ p(x) = x5₋3x2 + 2x✱ ♣♦✐sp(1) = 0✳

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

Pr♦♣♦s✐çã♦ ✶✳✹✳ ❙❡❥❛ p(x) ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ♥✉❧♦ t❛❧ q✉❡ ❣r❛✉(p(x)) _≥ 1✳ ❊♥tã♦✱

α_∈R s❡rá ✉♠❛ r❛✐③ ❞❡ _p(x) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ _(x−α) ❞✐✈✐❞✐r_p(x)✳

❉❡♠♦♥str❛çã♦✿ (_⇒) ❙✉♣♦♥❤❛♠♦s q✉❡ p(α) = 0✳ ❋❛③❡♥❞♦ ❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞❡

p(x)♣♦r (x₋α) ✭✈❡❥❛ ❚❡♦r❡♠❛ ✶✳✶✮✱ ♦❜t❡♠♦s

p(x) = (x₋α)h(x) +r(x),

♦♥❞❡ r(x)_≡0 ♦✉0_≤gr(r(x))<1✳

❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r r(x) = c_∈R ❡

p(x) = (x₋α)h(x) +c.

❆✈❛❧✐❛♥❞♦ p(x) ❡♠ α✱ t❡♠♦s

0 =p(α) = h(α)(α₋α) +c,

♦✉ s❡❥❛✱ r(x) = c= 0✱ ♦ q✉❡ ♠♦str❛ q✉❡(x₋α) ❞✐✈✐❞❡ p(x)✳

(_⇐)❙✉♣♦♥❤❛♠♦s q✉❡(x₋α)❞✐✈✐❞❛p(x)✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦h(x)t❛❧ q✉❡

p(x) =h(x)(x₋α)✳ ▲♦❣♦✱p(α) = h(α)(α₋α) = 0✳

❘❛✐③ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ✸✺

Pr♦♣♦s✐çã♦ ✶✳✺✳ ❙❡❥❛ p(x) ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ♥✉❧♦✳ ❙❡ p(x) t✐✈❡r ❣r❛✉ n✱ ❡♥tã♦ p(x)

t❡rá✱ ♥♦ ♠á①✐♠♦✱ n r❛í③❡s r❡❛✐s✳

❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ s❡rá ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡n =grau(p(x))✳

❙❡n= 0✱ ❡♥tã♦ p(x) =a0 6= 0 ♥ã♦ t❡rá r❛í③❡s r❡❛✐s ❡ ♦ r❡s✉❧t❛❞♦ s❡rá ✈á❧✐❞♦✳

❙✉♣♦♥❤❛♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈❡r❞❛❞❡✐r♦ ♣❛r❛ ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ n _≥ 0 ❡

❝♦♥s✐❞❡r❡♠♦s p(x) ✉♠ ♣♦❧✐♥ô♠✐♦ t❛❧ q✉❡ ❣r❛✉(p(x)) = n+ 1✳

❙❡p(x) ♥ã♦ t✐✈❡r r❛í③❡s ❡♠ R✱ ♥ã♦ t❡♠♦s ♥❛❞❛ ❛ ❞❡♠♦♥str❛r✳ ❊♥tã♦✱ ❞✐❣❛♠♦s q✉❡

p(x) t❡♥❤❛ ✉♠❛ r❛✐③ α _∈ R✳ P❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ _x−α ❞✐✈✐❞❡ _p(x)✳ ❉❛í✱ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ h(x)t❛❧ q✉❡ p(x) =h(x)(x₋α)✱ ♦♥❞❡ ❣r❛✉(h(x)) =n✳ P♦r ❤✐♣ót❡s❡ ❞❡

✐♥❞✉çã♦✱ h(x) t❡♠✱ ♥♦ ♠á①✐♠♦✱ n r❛í③❡s✳ ❖❜s❡r✈❡♠♦s q✉❡✿

β _∈R é r❛✐③ ❞❡ _{p(x) ⇐⇒ 0 =p(β) =h(β)(β−α)}

⇐⇒ h(β) = 0 ♦✉β =α

⇐⇒ β _∈R é r❛✐③ ❞❡ _h(x) ♦✉_{β =α.} P♦rt❛♥t♦✱ p(x) t❡♠✱ ♥♦ ♠á①✐♠♦✱ n+ 1 r❛í③❡s✳

Pr♦♣♦s✐çã♦ ✶✳✻✳ ❯♠ ♣♦❧✐♥ô♠✐♦p(x)s❡rá ❞✐✈✐sí✈❡❧ ♣♦r(x₋α₁)_{· · ·}(x₋αj)· · ·(x−αn)✱ ♦♥❞❡α₁,_{· · ·} , αj,· · · , αnsã♦ ♥ú♠❡r♦s r❡❛✐s ❞✐st✐♥t♦s s❡✱ ❡ s♦♠❡♥t❡ s❡✱α1,· · · , αj,· · · , αn ❢♦r❡♠ r❛í③❡s ❞✐st✐♥t❛s ❞❡ p(x)✳

❉❡♠♦♥str❛çã♦✿(=_⇒)❙❡(x₋α₁)_{· · ·}(x₋αj)· · ·(x−αn)❞✐✈✐❞✐rp(x)✱ ❡①✐st✐rá ✉♠ ♣♦❧✐♥ô♠✐♦ h(x)❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ R t❛❧ q✉❡

p(x) = (x₋α₁)(x₋α₂)_{· · ·}(x₋αj)· · ·(x−αn)h(x).

❉❡ss❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ q✉❡ p(α₁) =p(α₂) = _{· · ·}=p(αj) = · · ·=p(αn) = 0✳

▲♦❣♦✱α₁,_{· · ·} , αj,· · · , αnsã♦ r❛í③❡s r❡❛✐s ❞✐st✐♥t❛s ❞❡p(x)✱ ❥á q✉❡α1,· · ·, αj,· · · , αn sã♦ ♥ú♠❡r♦s r❡❛✐s ❞✐st✐♥t♦s✳

(_⇐=) ❙❡❥❛♠ α₁, α₂, ..., αn r❛í③❡s r❡❛✐s ❞✐st✐♥t❛s ❞❡ p(x)✳ ❈♦♠♦α₁ é r❛✐③ ❞❡ p(x)✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

p(x) = (x₋α1)q1(x).

❈♦♠♦α2 t❛♠❜é♠ é ✉♠❛ r❛✐③ ❞❡ p(x)✱ s✉❜st✐t✉✐♥❞♦ x ♣♦rα2 ♥❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r✱

♦❜t❡♠♦s✿

0 =p(α2) = (α2−α1)q1(α2).

❈♦♠♦(α2−α1)6= 0✱ ♣♦✐sα1 ❡ α2 sã♦ r❛í③❡s ❞✐st✐♥t❛s ❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s

r❡❛✐s é ③❡r♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✉♠ ❞♦s ❢❛t♦r❡s é ③❡r♦✱ ❞❡✈❡♠♦s t❡r q1(α2) = 0✳ P♦rt❛♥t♦✱

α2 é r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦ q1(x)✱ ❞♦♥❞❡ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ q1(x) = (x−α2)q2(x) ❡✱ ♣♦r

❝♦♥s❡❣✉✐♥t❡✱