O ENSINO DE POLINÔMIOS UTILIZANDO A HISTÓRIA DA MATEMÁTICA COMO RECURSO DIDÁTICO

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❘❯❘❆▲ ❉❊ P❊❘◆❆▼❇❯❈❖ ✲ ❯❋❘P❊ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ✲ P❘❖❋▼❆❚

❖ ❊◆❙■◆❖ ❉❊ P❖▲■◆Ô▼■❖❙ ❯❚■▲■❩❆◆❉❖ ❆

❍■❙❚Ó❘■❆ ❉❆ ▼❆❚❊▼➪❚■❈❆ ❈❖▼❖ ❘❊❈❯❘❙❖

❉■❉➪❚■❈❖

❋r❛♥❝✐s❝❛ ❆❧✈❡s ❞❡ ❙♦✉③❛

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❘❯❘❆▲ ❉❊ P❊❘◆❆▼❇❯❈❖ ✲ ❯❋❘P❊ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ✲ P❘❖❋▼❆❚

❖ ❊◆❙■◆❖ ❉❊ P❖▲■◆Ô▼■❖❙ ❯❚■▲■❩❆◆❉❖ ❆ ❍■❙❚Ó❘■❆ ❉❆ ▼❆❚❊▼➪❚■❈❆ ❈❖▼❖ ❘❊❈❯❘❙❖ ❉■❉➪❚■❈❖

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣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♦✿ ❊❞✉❝❛çã♦ ▼❛t❡♠át✐❝❛

❖r✐❡♥t❛❞♦r❛✿ ❉r❛✳ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛

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❋r❛♥❝✐s❝❛ ❆❧✈❡s ❞❡ ❙♦✉③❛

❖ ❊◆❙■◆❖ ❉❊ P❖▲■◆Ô▼■❖❙ ❯❚■▲■❩❆◆❉❖ ❆ ❍■❙❚Ó❘■❆ ❉❆ ▼❆❚❊▼➪❚■❈❆ ❈❖▼❖ ❘❊❈❯❘❙❖ ❉■❉➪❚■❈❖✴ ❋r❛♥❝✐s❝❛ ❆❧✈❡s ❞❡ ❙♦✉③❛✳ ✕ ❘❡❝✐❢❡✱ ▼❛rç♦ ❞❡ ✷✵✶✻

❄❄ ♣✳ ✐❧✳ ✭❛❧❣✉♠❛s ❝♦❧♦r✳✮ ❀ ✸✵ ❝♠✳

❖r✐❡♥t❛❞♦r❛✿ ❉r❛✳ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛

❉✐ss❡rt❛çã♦ ✕ ❯❋❘P❊✱ ▼❛rç♦ ❞❡ ✷✵✶✻✳

✶✳ P❛❧❛✈r❛✲❝❤❛✈❡✶✳ ✷✳ P❛❧❛✈r❛✲❝❤❛✈❡✷✳ ■✳ ❉r❛✳ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛✳ ■■✳ ❯❋❘P❊✳ ■■■✳ ❖ ❊◆❙■◆❖ ❉❊ P❖▲■◆Ô▼■❖❙ ❯❚■▲■❩❆◆❉❖ ❆ ❍■❙❚Ó❘■❆ ❉❆ ▼❆❚❊▼➪❚■❈❆ ❈❖▼❖ ❘❊❈❯❘❙❖ ❉■❉➪❚■❈❖

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❖ ❊◆❙■◆❖ ❉❊ P❖▲■◆Ô▼■❖❙ ❯❚■▲■❩❆◆❉❖ ❆

❍■❙❚Ó❘■❆ ❉❆ ▼❆❚❊▼➪❚■❈❆ ❈❖▼❖ ❘❊❈❯❘❙❖

❉■❉➪❚■❈❖

❋r❛♥❝✐s❝❛ ❆❧✈❡s ❞❡ ❙♦✉③❛

❉✐ss❡rt❛çã♦ ❥✉❧❣❛❞❛ ❛❞❡q✉❛❞❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❞❡❢❡♥✲ ❞✐❞❛ ❡ ❛♣r♦✈❛❞❛ ♣♦r ✉♥❛♥✐♠✐❞❛❞❡ ❡♠ ✴ ✴✷✵✶✻ ♣❡❧❛ ❝♦♠✐ssã♦ ❡①❛♠✐♥❛❞♦r❛✳

❖r✐❡♥t❛❞♦r❛✿

❉r❛✳ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛

❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

❉r✳ ❘♦ss ❆❧✈❡s ❞♦ ◆❛s❝✐♠❡♥t♦ ❯❋❘P❊

❉r✳ P❛✉❧♦ ❋✐❣✉❡✐r❡❞♦ ▲✐♠❛ ❯❋P❊

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛ ♠✐♥❤❛ ♠ã❡ ♣♦r s❡r ✉♠ ❡①❡♠♣❧♦ ❞❡ ❤♦♥❡st✐❞❛❞❡✱ ❞❡ ♠✉❧❤❡r✱ ❞❡ ❝♦r❛❣❡♠ ❡ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✳

❆ ♠✐♥❤❛ ✜❧❤❛ ❨❛s♠✐♠✱ ♣♦r s❡r ♠❡✉ ♣♦rt♦ s❡❣✉r♦✱ ♣♦✐s ♦s s❡✉s ♦❧❤♦s ♠❡ tr❛♥s♠✐t❡♠ tr❛♥q✉✐❧✐❞❛❞❡ ❡ ♣❛③✳

❆♦ ♠❡✉ ✜❧❤♦ ❨❛❣♦✱ ♣♦r s❡r ♦ ♠♦t✐✈♦ ❞❛ ♠✐♥❤❛ ❢♦rç❛ ❡ ♦❜st✐♥❛çã♦✳

❆♦ ♠❡✉ ❝♦♠♣❛♥❤❡✐r♦ ❞❡ ✈✐❞❛ ❡ ❛♠✐❣♦ ❈❛r❧♦s✱ ♣♦r ♠❡ ❛♣♦✐❛r ❡ ♥ã♦ ❞❡✐①❛r q✉❡ ♦ ❞♦♠♦✱ ♦ q✉❛❧ ❡✉ ❝r✐❡✐ ❛♦ ♠❡✉ r❡❞♦r✱ ♦ ✐♠♣❡❞✐ss❡ ❞❡ ✈❡r ❛ ♠✐♥❤❛ ✈❡r❞❛❞❡✐r❛ ❡ssê♥❝✐❛✳

❆ ♠✐♥❤❛ ♠❛❞r✐♥❤❛✱ ❝❛r✐♥❤♦s❛♠❡♥t❡ ❝❤❛♠❛❞❛ ❞❡ ❉✐♥❞✐♥❤❛✱ ♣♦r t❡r ♠❡ ❞❛❞♦ s❡✉ ❛♠♦r ❡ ♠❡ ❡♥s✐♥❛❞♦ ♦ ❝❛♠✐♥❤♦ ❞❛ ❢é ❛té ♦s ú❧t✐♠♦s ✐♥st❛♥t❡s ❞❛ s✉❛ ✈✐❞❛✳ ■ ✇✐❧❧ ❧♦✈❡ ②♦✉ ❢♦r❡✈❡r✳

❆ ♠✐♥❤❛ ❜✐s❛✈ó✱ ♣♦r ♠❡ ❛♠❛r✱ ♠❡s♠♦ q✉❡ ❛s ✈❡③❡s s❡ ❡sq✉❡❝❡ss❡ q✉❡♠ ❡✉ ❡r❛✳ ❆ ❙❇▼ ♣♦r t❡r ♠❡ ♣r♦♣♦r❝✐♦♥❛❞♦ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❝✉rs❛r ♦ ♠❡str❛❞♦ ❞♦ P❘❖❋✲ ▼❆❚✳

❆ ❈❆P❊❙ ♣♦r t❡r ♠❡ ❞❛❞♦ r❡❝✉rs♦s ♣❛r❛ ❝✉rs❛r ♦ P❘❖❋▼❆❚✳

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◆✉♥❝❛ ✉♠ ❢❡♥ô♠❡♥♦ ❤✐stór✐❝♦ s❡ ❡①♣❧✐❝❛ ♣❧❡✲ ♥❛♠❡♥t❡ ❢♦r❛ ❞♦ ❡st✉❞♦ ❞♦ s❡✉ ♠♦♠❡♥t♦✳ ❊ ✐st♦ é ✈á❧✐❞♦ ♣❛r❛ t♦❞❛s ❛s ❡t❛♣❛s ❞❛ ❡✈♦❧✉✲ çã♦✳ P❛r❛ ❛q✉❡❧❛ ❡♠ q✉❡ ✈✐✈❡♠♦s✱ ❝♦♠♦ ♣❛r❛ ♦✉tr❛s✳ ❏á ✉♠ ♣r♦✈ér❜✐♦ ár❛❜❡ ❞✐ss❡r❛✿ ✏♦s ❤♦♠❡♥s ♣❛r❡❝❡♠✲s❡ ♠❛✐s ❝♦♠ ♦ s❡✉ t❡♠♣♦ q✉❡ ❝♦♠ ♦s s❡✉s ♣❛✐s✑✳

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❘❡s✉♠♦

❖s ♣♦❧✐♥ô♠✐♦s sã♦ ❞❡ s✉♠❛ r❡❧❡✈â♥❝✐❛ ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛ ❡ q✉❛♥❞♦ ❛ss♦❝✐❛❞♦s ❛ ❢✉♥çõ❡s ♠♦❞❡❧❛♠ ✈ár✐♦s ❢❡♥ô♠❡♥♦s ❞♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✳ ❊ss❡ ❛ss✉♥t♦ é ❛❜♦r❞❛❞♦✱ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✱ ♥♦ ✽➸ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♣♦ré♠ ♦s ❛❧✉♥♦s ❝❤❡❣❛♠ ❛♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ s✉♣❡r✐♦r ❝♦♠ ✈ár✐❛s ❞✐✜❝✉❧❞❛❞❡s ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞♦ ♠❡s♠♦✳ ❊ss❛s ❞✐✜❝✉❧❞❛❞❡s ♦❝♦rr❡♠ ♣♦r ✐♥ú♠❡r♦s ♠♦t✐✈♦s✱ s❡♥❞♦ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s ❛ ❛✈❡rsã♦ q✉❡ ♦s ❛❧✉♥♦s t❡♠ ♣❡❧❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛ ❡ ♣♦r ❡ss❛ r❛③ã♦ ❡st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ✉♠ ❞♦s s❡✉s ♦❜❥❡t✐✈♦s à ❛♣r♦♣r✐❛çã♦ ❞❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ r❡❝✉rs♦ ❞✐❞át✐❝♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ♣♦❧✐♥ô♠✐♦s✳ ❖ ♦✉tr♦ ♦❜❥❡t✐✈♦ é ❡❧❛❜♦r❛r ✉♠❛ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ❞✐❞át✐❝♦ ♣❛r❛ s❡r ✉t✐❧✐③❛❞♦ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✳ P❛r❛ ✈❡r✐✜❝❛r ❛ ❡✜❝á❝✐❛ ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ r❡❝✉rs♦ ❞✐❞át✐❝♦ ♥♦ ❡♥s✐♥♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❢♦✐ r❡❛❧✐③❛❞❛ ✉♠❛ ♣❡sq✉✐s❛ ❝♦♠ ❛❧✉♥♦s ❞❛ ✶➟ sér✐❡ ❞♦ ❝✉rs♦ té❝♥✐❝♦ ❡♠ ✐♥❢♦r♠át✐❝❛ ♣❛r❛ ✐♥t❡r♥❡t ✐♥t❡❣r❛❞♦ ❛♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞♦ ■❋❈❊✲❈❛♠♣✉s ❈r❛t♦✱ ❛♣❧✐❝❛♥❞♦ ❛ s❡❣✉✐♥t❡ ♠❡t♦❞♦❧♦❣✐❛✿ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ t❡st❡✱ ♦ q✉❛❧ ❢♦✐ ❝❤❛♠❛❞♦ ❞❡ ❚❡st❡ ✶✱ ♣❛r❛ ✈❡r✐✜❝❛r ❛s ❞✐✜❝✉❧❞❛❞❡s ❞♦s ❛❧✉♥♦s ♥❛ r❡s♦❧✉çã♦ ❞❡ q✉❡stõ❡s ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❞❡♣♦✐s ❢♦✐ r❡❛❧✐③❛❞❛ ✉♠❛ ♦✜❝✐♥❛ ♣❛r❛ ❡st✉❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✉s❛♥❞♦ ❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ r❡❝✉rs♦ ❞✐❞át✐❝♦ ❡ ♣❛r❛ ✜♥❛❧✐③❛r ❢♦✐ ❛♣❧✐❝❛❞♦ ✉♠ s❡❣✉♥❞♦ t❡st❡ ✭❚❡st❡ ✷✮ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ❤♦✉✈❡ ❛❧❣✉♠ ❛✈❛♥ç♦ ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞♦s ❛❧✉♥♦s✳

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❆❜str❛❝t

P♦❧②♥♦♠✐❛❧s ❛r❡ ♦❢ ♣❛r❛♠♦✉♥t ✐♠♣♦rt❛♥❝❡ ❢♦r ♠❛t❤❡♠❛t✐❝s ❛♥❞ ❢✉♥❝t✐♦♥s ✇❤❡♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ♠♦❞❡❧✐♥❣ ✈❛r✐♦✉s ♣❤❡♥♦♠❡♥❛ ♦❢ ♦✉r ❞❛② t♦ ❞❛②✳ ❚❤✐s s✉❜❥❡❝t ✐s ❛❞❞r❡ss❡❞ ❢♦r t❤❡ ✜rst t✐♠❡ ✐♥ ✽ ②❡❛rs ♦❢ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧✱ ❜✉t st✉❞❡♥ts ❛rr✐✈❡ ❛t s❡❝♦♥❞❛r② ❛♥❞ ❤✐❣❤❡r ❡❞✉❝❛t✐♦♥ ✇✐t❤ ♠✉❧t✐♣❧❡ ❧❡❛r♥✐♥❣ ❞✐✣❝✉❧t✐❡s ♦❢ ✐t✳ ❚❤❡s❡ ❞✐✣❝✉❧t✐❡s ♦❝❝✉r ❢♦r ♠❛♥② r❡❛✲ s♦♥s✱ t❤❡ ♠❛✐♥ ♦♥❡ ❜❡✐♥❣ t❤❡ ❛✈❡rs✐♦♥ t❤❛t st✉❞❡♥ts ❤❛✈❡ ❜② ♠❛t❤ ❝❧❛ss❡s ❛♥❞ ❢♦r t❤❛t r❡❛s♦♥ t❤✐s ✇♦r❦ ❤❛s ❛s ♦♥❡ ♦❢ ✐ts ♦❜❥❡❝t✐✈❡s t♦ t❤❡ ❤✐st♦r② ♦❢ ♠❛t❤❡♠❛t✐❝s ♦✇♥❡rs❤✐♣ ❛s ❛ t❡❛❝❤✐♥❣ r❡s♦✉r❝❡ ❢♦r t❡❛❝❤✐♥❣ ❛♥❞ ❧❡❛r♥✐♥❣ ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ♦t❤❡r ❣♦❛❧ ✐s t♦ ❞❡✈❡❧♦♣ ❛ t❡❛❝❤✐♥❣ s✉♣♣♦rt ♠❛t❡r✐❛❧ ❢♦r ✉s❡ ✐♥ t❤❡ ❝❧❛ssr♦♦♠✳ ❚♦ ❝❤❡❝❦ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ ✉s❡ ♦❢ t❤❡ ❤✐st♦r② ♦❢ ♠❛t❤❡♠❛t✐❝s ❛s ❛ t❡❛❝❤✐♥❣ r❡s♦✉r❝❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t❡❛❝❤✐♥❣ ❛ s✉r✲ ✈❡② ✇❛s ❝♦♥❞✉❝t❡❞ ✇✐t❤ st✉❞❡♥ts ❢r♦♠ t❤❡ ✶st s❡r✐❡s ♦❢ t❡❝❤♥✐❝❛❧ ❝♦✉rs❡ ✐♥ ✐♥❢♦r♠❛t✐❝s ❢♦r ✐♥t❡❣r❛t❡❞ ✐♥t❡r♥❡t t♦ ❤✐❣❤ s❝❤♦♦❧ t❤❡ ■❋❈❊✲❈❛♠♣✉s ❈r❛t♦✱ ❛♣♣❧②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞♦❧♦❣②✿ ❆♣♣❧✐❝❛t✐♦♥ ❛ t❡st ✇❤✐❝❤ ✇❛s ❝❛❧❧❡❞ ❚❡st ✶✱ t♦ ❝❤❡❝❦ st✉❞❡♥ts✬ ❞✐✣❝✉❧t✐❡s ✐♥ r❡s♦❧✈✐♥❣ ✐ss✉❡s ✐♥✈♦❧✈✐♥❣ t❤❡ ❝♦♥❝❡♣t ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ t❤❡♥ ❤❡❧❞ ❛ ✇♦r❦s❤♦♣ ❢♦r ♣♦❧②♥♦✲ ♠✐❛❧s st✉❞② ✉s✐♥❣ t❤❡ ♠❛t❤❡♠❛t✐❝s ♦❢ ❤✐st♦r② ❛s ❛ t❡❛❝❤✐♥❣ t♦♦❧ ❛♥❞ t♦ ✜♥✐s❤ ✇❡ ❛♣♣❧✐❡❞ ❛ s❡❝♦♥❞ t❡st ✭❚❡st ✷✮ t♦ s❡❡ ✐❢ t❤❡r❡ ✇❛s s♦♠❡ ♣r♦❣r❡ss ✐♥ st✉❞❡♥t ❧❡❛r♥✐♥❣✳

(16)

(17)

▲✐st❛ ❞❡ ✐❧✉str❛çõ❡s

(18)

(19)

▲✐st❛ ❞❡ t❛❜❡❧❛s

(20)

(21)

❙✉♠ár✐♦

(22)

✷✳✻ Pr♦❞✉t♦s ◆♦tá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✻✳✶ ◗✉❛❞r❛❞♦ ❞❛ ❙♦♠❛ ❞❡ ❉♦✐s ❚❡r♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✻✳✷ ◗✉❛❞r❛❞♦ ❞❛ ❉✐❢❡r❡♥ç❛ ❞❡ ❉♦✐s ❚❡r♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✷✳✻✳✸ Pr♦❞✉t♦ ❞❛ ❙♦♠❛ ♣❡❧❛ ❉✐❢❡r❡♥ç❛ ❞❡ ❉♦✐s ❚❡r♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✷✳✻✳✹ ❈✉❜♦ ❞❛ ❙♦♠❛ ❡ ❞❛ ❉✐❢❡r❡♥ç❛ ❞❡ ❉♦✐s ❚❡r♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✷✳✻✳✺ ❋❛t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✷✳✻✳✻ ❈♦♠❜✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✷✳✻✳✼ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✷✳✻✳✽ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✷✳✻✳✾ ❋❛t♦r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✷✳✼ ❊①❡r❝í❝✐♦s ♣r♦♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸ ❚❡st❡ ✷ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✸✳✶ ❆♥á❧✐s❡ ❞❛s q✉❡stõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✸✳✶✳✶ ◗✉❡stã♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✸✳✶✳✷ ◗✉❡stã♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✸✳✶✳✸ ◗✉❡stã♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✸✳✶✳✹ ◗✉❡stã♦ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✸✳✶✳✺ ◗✉❡stã♦ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✶✳✻ ◗✉❡stã♦ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✶✳✼ ◗✉❡stã♦ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✸✳✶✳✽ ◗✉❡stã♦ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✸✳✶✳✾ ◗✉❡stã♦ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✸✳✶✳✶✵ ◗✉❡stã♦ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✸✳✶✳✶✶ ◗✉❡stã♦ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✸✳✶✳✶✷ ◗✉❡stã♦ ✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✸✳✶✳✶✸ ◗✉❡stã♦ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✸✳✶✳✶✹ ◗✉❡stã♦ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✸✳✶✳✶✺ ◗✉❡stã♦ ✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✸✳✷ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✸✳✸ ❆♥á❧✐s❡ ❝♦♠♣❛r❛t✐✈❛ ❞♦s r❡s✉❧t❛❞♦s ♣♦r q✉❡stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✹ ❆♥á❧✐s❡ ❝♦♠♣❛r❛t✐✈❛ ❞♦s r❡s✉❧t❛❞♦s ♣♦r ♥♦t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾

(23)

❆✳✹ P❧❛♥♦ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ❆✳✺ P❧❛♥♦ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ❆◆❊❳❖ ❇ ▼❛t❡r✐❛✐s ✉t✐❧✐③❛❞♦s ♥❛s ❛✉❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸

(24)

(25)

✷✸

■♥tr♦❞✉çã♦

❆t✉❛❧♠❡♥t❡✱ ♥♦s ❝✉rs♦s ❞❡ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛✱ ❡stá ❤❛✈❡♥❞♦ ✉♠❛ ♣r❡♦✲ ❝✉♣❛çã♦ ♠❛✐s ❛❝❡♥t✉❛❞❛ ❝♦♠ ♦ ❡st✉❞♦ ❞❡ ❢♦r♠❛s ❡ r❡❝✉rs♦s q✉❡ ♣♦❞❡♠ ♠❡❧❤♦r❛r ♦ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛✱ ❡ é r❡❛❧♠❡♥t❡ ♥❡ss❛ ❢❛s❡ q✉❡ ❞❡✈❡ s❡r ♣r♦♣✐❝✐❛❞♦ ❛♦s ❢✉t✉r♦s ❞♦❝❡♥t❡s ❛ ♦♣♦rt✉♥✐❞❛❞❡

(. . .)❞❡ tr❛❜❛❧❤❛r s❡❣✉♥❞♦ ♠❡t♦❞♦❧♦❣✐❛s ❞❡ ❡♥s✐♥♦ ❡ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞✐✲

✈❡rs✐✜❝❛❞❛s✱ ❞❡ ♠♦❞♦ ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❞❡ ❝❛♣❛❝✐❞❛❞❡s✱ ❞❡ ❛t✐t✉❞❡s ❡ ❞❡ ✈❛❧♦r❡s✳ ❊st❛ ❡①♣♦s✐çã♦ ❛ ❞✐❢❡r❡♥t❡s ♠ét♦✲ ❞♦s t❛♠❜é♠ ❢✉♥❝✐♦♥❛ ❝♦♠♦ ✉♠ ♠❡❝❛♥✐s♠♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✳ ✭P❖◆❚❊✱ ✷✵✵✵✱ ♣✳ ✶✺✮

❙❡ ❢❛③ ♥❡❝❡ssár✐♦ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ❤✐stór✐❛ ❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❢❛ç❛♠ ♣❛rt❡ ❞❛ ❢♦r♠❛çã♦ ❞♦s ♣r♦❢❡ss♦r❡s ♣❛r❛ q✉❡ ♦s ♠❡s♠♦s t❡♥❤❛♠ ❡❧❡♠❡♥t♦s q✉❡ ❧❤❡s ♣❡r✲ ♠✐t❛♠ ♠♦str❛r ❛♦s ❞✐s❝❡♥t❡s ❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠❛ ❝✐ê♥❝✐❛ ❞✐♥â♠✐❝❛ s❡♠♣r❡ ❛❜❡rt❛ à ✐♥s❡rçã♦ ❞❡ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s✳

❆ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛ é ✉♠ r❡❝✉rs♦ q✉❡ ♣♦❞❡ ❡ ❞❡✈❡ s❡r ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❞♦✲ ❝❡♥t❡s ❞❡ ♠❛t❡♠át✐❝❛✱ ❥á q✉❡ ❛ ♠❡s♠❛ é ❛♣♦♥t❛❞❛ ♣♦r ✈ár✐♦s ♣❡sq✉✐s❛❞♦r❡s✱ t❛✐s ❝♦♠♦ ❯❜✐r❛t❛♥ ❉✬❆♠❜r♦s✐♦✱ ❆♥t♦♥✐♦ ▼✐❣✉❡❧✱ ▼❛r✐❛ ➶♥❣❡❧❛ ▼✐♦r✐♠✱ ❆r❧❡t❡ ❞❡ ❏❡s✉s ❇r✐t♦✱ ❙ér✲ ❣✐♦ ❘♦❜❡rt♦ ◆♦❜r❡✱ ❘♦s❛ ▲ú❝✐❛ ❙✈❡r③✉t ❇❛r♦♥✐ ❡ ❉✐r❦ ❏❛♥ ❙tr✉✐❦✱ ❝♦♠♦ ✉♠ ✐♥str✉♠❡♥t♦ ❞✐❞át✐❝♦ ✐♠♣♦rt❛♥t❡ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ♠❛t❡♠át✐❝❛✱ ♣♦✐s ♣♦❞❡ tr❛✲ ③❡r ✐♥ú♠❡r❛s ❝♦♥tr✐❜✉✐çõ❡s t❛♥t♦ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ♣❛r❛ ♦s ❛❧✉♥♦s ❞❡ss❛ ❞✐s❝✐♣❧✐♥❛✳

P♦r t♦❞♦s ♦s ♠♦t✐✈♦s ❝✐t❛❞♦s ❛❝✐♠❛ ❡ t❛♠❜é♠ ♣♦r t♦r♥❛r ✉♠❛ ❛✉❧❛ ♠❛✐s ♣r❛③❡r♦s❛ é q✉❡ ❡ss❡ tr❛❜❛❧❤♦ ✉t✐❧✐③❛ ❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ r❡❝✉rs♦ ❞✐❞át✐❝♦ ♥♦ ❡st✉❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♦❜❥❡t✐✈❛♥❞♦ ❛ r❡❣r❡ssã♦ ❞❛ ❛✈❡rsã♦ q✉❡ ♦s ❞✐s❝❡♥t❡s t❡♠ ♣❡❧❛ ♠❛t❡♠át✐❝❛✱ ❡ ❝♦♥❢♦r♠❡ ♦ P❈◆✰✿

❆ ♠❛t❡♠át✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❡✈❡ s❡r ❝♦♠♣r❡❡♥❞✐❞❛ ❝♦♠♦ ✉♠❛ ♣❛r❝❡❧❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❤✉♠❛♥♦ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❡ t♦❞♦s ♦s ❥♦✈❡♥s✱ q✉❡ ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ✈✐sã♦ ❞❡ ♠✉♥❞♦✱ ♣❛r❛ ❧❡r ❡ ✐♥t❡r♣r❡t❛r ❛ r❡❛❧✐❞❛❞❡ ❡ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❝❛♣❛❝✐❞❛❞❡s q✉❡ ❞❡❧❡s s❡rã♦ ❡①✐❣✐❞❛s ❛♦ ❧♦♥❞♦ ❞❛ ✈✐❞❛ s♦❝✐❛❧ ❡ ♣r♦✜ss✐♦♥❛❧✳ ✭P❈◆✰✱ ♣✳ ✶✶✶✮

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✷✹ ■♥tr♦❞✉çã♦

t❛r ❡♠ ✉♠ ❜✐♠❡str❡ ❝♦♠ ❛s ♠❛✐♦r❡s ♥♦t❛s ❡ s❡♠ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❛♣❧✐❝❛r ❛ r❡❝✉♣❡r❛çã♦ ❜✐♠❡str❛❧✱ ♣♦ré♠ ♥ã♦ é ♦ q✉❡ ♦❝♦rr❡ ♥❛ ♣rát✐❝❛✳

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✷✺

✶ ❚❡st❡ ✶ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

❖ ❚❡st❡ ✶ ❢♦✐ ❛♣❧✐❝❛❞♦ ♣❛r❛ ✷✾ ❛❧✉♥♦s ❞❛ ✶➟ sér✐❡ ❞♦ ❝✉rs♦ ❚é❝♥✐❝♦ ❡♠ ■♥❢♦r♠át✐❝❛ ♣❛r❛ ■♥t❡r♥❡t ■♥t❡❣r❛❞♦ ❛♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞♦ ■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞❡ ❊❞✉❝❛çã♦✱ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛✱ ❧♦❝❛❧✐③❛❞♦ ♥♦ ❡st❛❞♦ ❞♦ ❈❡❛rá ♥❛ ❝✐❞❛❞❡ ❞❡ ❈r❛t♦✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ t❡st❡ ❡r❛ ❛✈❡r✐❣✉❛r s❡ ♦s ❞✐s❝❡♥t❡s ❛✐♥❞❛ t✐♥❤❛♠ ❞✐✜❝✉❧❞❛❞❡s ♣❛r❛ r❡s♦❧✈❡r❡♠ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♣♦✐s t❡♦r✐❝❛♠❡♥t❡ ❡❧❡s ❥á ❞❡✈❡r✐❛♠ t❡r s❡ ❛♣r♦♣r✐❛❞♦ ❞❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦✱ ✈✐st♦ q✉❡ ❡ss❡ ❛ss✉♥t♦ é ♠✐♥✐str❛❞♦ ♥♦ ✽➸ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✳ ❖ t❡st❡ ❢♦✐ ❝♦♠♣♦st♦ ♣♦r ✶✺ q✉❡stõ❡s✱ ❛s q✉❛✐s ❛❜♦r❞❛♠ ❛ ❞❡✜♥✐çã♦✱ ✈❛❧♦r ♥✉♠ér✐❝♦✱ ❣r❛✉✱ ♣r♦❞✉t♦ ♥♦tá✈❡✐s✱ ❛♣❧✐❝❛çã♦✱ ♦♣❡r❛çõ❡s✱ r❛í③❡s ❞❡ ❡q✉❛çõ❡s ❡ ❢❛t♦r❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✳ ❆❜❛✐①♦ s❡rá ♠♦str❛❞♦ ❛ ❛♥á❧✐s❡ ❞❛s r❡s♣♦st❛s ❞♦s ❞✐s❝❡♥t❡s ❡♠ ❝❛❞❛ q✉❡stã♦ ❡ ❞❡♣♦✐s ✉♠❛ ❛♥á❧✐s❡ ♠❛✐s ❣❡r❛❧ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳

✶✳✶ ❆♥á❧✐s❡ ❞❛s q✉❡stõ❡s

✶✳✶✳✶ ◗✉❡stã♦ ✶

❆ss✐♥❛❧❡ ❛ ❛❧t❡r♥❛t✐✈❛ q✉❡ r❡♣r❡s❡♥t❛ ✉♠ ♣♦❧✐♥ô♠✐♦ ♥❛ ✈❛r✐á✈❡❧x✿ ✭❛✮ P(x) = x−3+ 5x ✭❝✮ P(x) = x(

1 2)

−1

+ 6x+ 1

✭❜✮ P(x) = x√2

+ 1 ✭❞✮ P(x) = x(21) −1

+ 8x+ 2

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈

❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭❞❡ ✉♠ t♦t❛❧ ❞❡ ✷✾ ♣❛rt✐❝✐♣❛♥t❡s ❝♦♥❢♦r♠❡ ❢♦✐ ❡s♣❡❝✐✜❝❛❞♦ ❛❝✐♠❛✮ q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✶✳

❚❛❜❡❧❛ ✶ ✕ ◗✉❡stã♦ ✶❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✹✹✱✾

❜ ✶✸✱✽

❝ ✸✶✱✵

❞ ✸✱✹

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✻✱✾

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✷✻ ❈❛♣ít✉❧♦ ✶✳ ❚❡st❡ ✶ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

✸✶✱✵✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✳ ❯♠ ❢❛t♦ ✐♠♣♦rt❛♥t❡ é q✉❡ ❞♦✐s ❛❧✉♥♦s ♥ã♦ ❛ss✐✲ ♥❛❧❛r❛♠ ♥❡♥❤✉♠❛ ❞❛s ❛❧t❡r♥❛t✐✈❛s✱ ❞❡✐①❛♥❞♦ ♦s s❡❣✉✐♥t❡s q✉❡st✐♦♥❛♠❡♥t♦s✿ ❊❧❡ ♥ã♦ s❛❜✐❛ ❡ ♣♦r ✐ss♦ ♥ã♦ ❛ss✐♥❛❧♦✉ ♦✉ ❡st❛✈❛ ❝♦♠ ♣r❡ss❛ ♣❛r❛ ❡♥tr❡❣❛r❄ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❡ss❡s ❛❧✉♥♦s ❛♦ r❡s♣♦♥❞❡r❡♠ ❛ ♣❡r❣✉♥t❛✿ ◗✉❛✐s sã♦ s❡✉s s❡♥t✐♠❡♥t♦s ❡♠ r❡❧❛çã♦ ❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛❄ ❢❡✐t❛ ♥♦ ♣r✐♠❡✐r♦ ❞✐❛ ❞❡ ❛✉❧❛ ❞❛ ♦✜❝✐♥❛✱ r❡s♣♦♥❞❡r❛♠ q✉❡ ♥ã♦ ❣♦st❛✈❛ ♦✉ t✐♥❤❛ s♦♥♦ ❞✉r❛♥t❡ ❛s ❛✉❧❛s ❡ ♣♦r ✐♥❝rí✈❡❧ q✉❡ ♣❛r❡ç❛ ✉♠ ❞♦s ♠❡s♠♦s só ❢❛❧t♦✉ ❛s ❛✉❧❛s ❞♦ ❞✐❛ ✷✷✴✵✶ ❡ ❞✉r❛♥t❡ ❛s ❛✉❧❛s ♣❛r❡❝✐❛ ❡st❛ s❡ s❡♥t✐♥❞♦ ❜❡♠ ❡ ♥ã♦ ❞♦r♠✐✉✱ ♦ q✉❡ ❧❡✈❛ ❛ ❝♦♥st❛t❛r q✉❡ ♦ r❡❝✉rs♦ ✉t✐❧✐③❛❞♦ s✉rt✐✉ ❡❢❡✐t♦ ♣♦s✐t✐✈♦✳ ❙❡ ❢❛③ ♥❡❝❡ssár✐♦ r❡ss❛❧t❛r q✉❡ ❞✐❛ ✷✷✴✵✶ ❛❧é♠ ❞❡ s❡r ✉♠❛ s❡①t❛✲❢❡✐r❛ ✭❡❧❡s ♥ã♦ t❡♠ ❛✉❧❛ ♥❛ s❡①t❛✮✱ ❝❤♦✈✐❛ ❜❛st❛♥t❡ ❡ ♥♦ss❛ ❡s❝♦❧❛ é ❧♦❝❛❧✐③❛❞❛ ♥♦ sít✐♦ ❆❧♠❡❝❡❣❛s✱ ♦ q✉❡ ❞✐✜❝✉❧t❛ ❛✐♥❞❛ ♠❛✐s ♦ ❛❝❡ss♦✳

✶✳✶✳✷ ◗✉❡stã♦ ✷

❙❡♥❞♦ Q(x) = ₋x2

−5x+ 6✱ ❡♥tã♦ ♦ ✈❛❧♦r ♥✉♠ér✐❝♦ ❞❡ Q(x)q✉❛♥❞♦ x ❢♦r ✐❣✉❛❧ ❛ ₋2é✿

✭❛✮ 12 ✭❜✮ 20 ✭❝✮ 0 ✭❞✮ 16

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❆

❆tr❛✈és t❛❜❡❧❛ ❛❜❛✐①♦ ♣♦❞❡✲s❡ ✈❡r ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✷✳

❚❛❜❡❧❛ ✷ ✕ ◗✉❡stã♦ ✷❚✶ ❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✸✹✱✺

❜ ✹✹✱✽

❝ ✷✵✱✼

❞ ✵

(29)

✶✳✶✳ ❆♥á❧✐s❡ ❞❛s q✉❡stõ❡s ✷✼

❈♦♠ ❜❛s❡ ♥♦ ❝á❧❝✉❧♦ ❛❝✐♠❛ ♦❜s❡r✈❛✲s❡ ❛ ❢❛❧t❛ ❞❡ ❛t❡♥çã♦ ❞❡ss❡s ❛❧✉♥♦s ♥♦ ♠♦♠❡♥t♦ ❞❡ r❡s♦❧✈❡r ✉♠❛ ❡①♣r❡ssã♦ ♥✉♠ér✐❝❛✳

✶✳✶✳✸ ◗✉❡stã♦ ✸

❙❡❥❛P(x) = 0 ♦ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✱ ❡♥tã♦ ♦ ❣r❛✉ ❞❡ P(x) é✿

✭❛✮ 0 ✭❜✮ 1 ✭❝✮ 2 ✭❞✮ ◆❡♥❤✉♠❛ ❞❛s r❡s♣♦st❛s ❛♥t❡r✐♦r❡s✳

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❉

❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛✲ t✐✈❛ ❞❛ q✉❡stã♦ ✸✳

❚❛❜❡❧❛ ✸ ✕ ◗✉❡stã♦ ✸❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✸✶✱✵

❜ ✶✼✱✷

❝ ✷✹✱✶

❞ ✷✵✱✼

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✼✱✵

❖ ♦❜❥❡t✐✈♦ ❞❡ss❛ q✉❡stã♦ ❡r❛ ✈❡r✐✜❝❛r s❡ ♦ ❞✐s❝❡♥t❡ ❝♦♠♣r❡❡♥❞✐❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❣r❛✉ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦✳ P♦ré♠ ❛♥❛❧✐s❛♥❞♦ ❛ ❚❛❜❡❧❛ ✸ ❝♦♥str✉í❞❛ ❛❝✐♠❛✱ ♦❜s❡r✈❛✲s❡ q✉❡ ✼✾✱✸✪ ❞♦s ❛❧✉♥♦s ♥ã♦ s❡ ❛♣r♦♣r✐❛r❛♠ ❞❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♣♦✐s s♦♠❡♥t❡ ✷✵✱✼✪ ❞♦s ❞✐s❝❡♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✱ q✉❡ é ❛ ❉✳ P♦ré♠ ❛ ♠❛✐♦r✐❛ ✭✸✶✱✵ ✪✮ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❆✱ ♦ q✉❡ ♠♦str❛ q✉❡ ❡❧❡s ❝♦♥❢✉♥❞❡♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❝♦♠ ♦ ♣ró♣r✐♦ ♣♦❧✐♥ô♠✐♦✳

✶✳✶✳✹ ◗✉❡stã♦ ✹

❈♦♥s✐❞❡r❡ ♦s ♣♦❧✐♥ô♠✐♦sP(x) = 4x3

−3x2

+ 4x+ 5❡Q(x) = 2x2

−5x₋2✳ ❙❛❜❡♥❞♦ K(x)

é ♦ ♣♦❧✐♥ô♠✐♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡P(x) ❡Q(x)✱ ❡♥tã♦ ♦ ❣r❛✉ ❞❡ K(x) é✿

✭❛✮ 2 ✭❜✮ 3 ✭❝✮ 5 ✭❞✮ 6

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈

(30)

✷✽ ❈❛♣ít✉❧♦ ✶✳ ❚❡st❡ ✶ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

❚❛❜❡❧❛ ✹ ✕ ◗✉❡stã♦ ✹❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✶✵✱✸

❜ ✷✼✱✻

❝ ✸✶✱✵

❞ ✷✹✱✶

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✼✱✵

❊ss❛ q✉❡stã♦ ✈✐s❛✈❛ ✈❡r✐✜❝❛r s❡ ♦ ❛❧✉♥♦ s❛❜✐❛ ♠✉❧t✐♣❧✐❝❛r ♣♦❧✐♥ô♠✐♦s✳ ❈♦♠ ❜❛s❡ ♥❛ t❛❜❡❧❛ ❚❛❜❡❧❛ ✹ ♦❜s❡r✈❛✲s❡ q✉❡ ✷✼✱✻✪ ❞♦s ❛❧✉♥♦s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❉✱ ♦ q✉❡ ❧❡✈❛ ❛ ❛❝r❡❞✐t❛r q✉❡ ❡❧❡s ❛❝❤❛♠ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ♣r❡s❡r✈❛ ♦ ♠❛✐♦r ❣r❛✉ ❞♦s ♣♦❧✐♥ô♠✐♦s ❡♥✈♦❧✈✐❞♦s ❡ ✸✶✱✵✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❈✱ q✉❡ ❡r❛ ❛ ❝♦rr❡t❛✱ ♠❛s s♦♠❡♥t❡ ✶✸✱✽✪ ❞❡ss❡s ❛❧✉♥♦s ✜③❡r❛♠ ❛❧❣✉♠ t✐♣♦ ❞❡ ❝á❧❝✉❧♦✳ ❯♠ ❡rr♦ q✉❡ ❢♦✐ ✈❡r✐✜❝❛❞♦✱ t❛❧✈❡③ ♣♦r ❢❛❧t❛ ❞❡ ❛t❡♥çã♦ ♥❛ ❧❡✐t✉r❛ ❞♦ ❡♥✉♥❝✐❛❞♦ ♦✉ ♣♦r ♥ã♦ s❛❜❡r❡♠ ❧❡r ❡ ✐♥t❡r♣r❡t❛r t❡①t♦s✱ ❢♦✐ ♦ ❝á❧❝✉❧♦ ❞❡ r❛í③❡s ❞❡ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛✐s s❡♠ ❛ q✉❡stã♦ ♣❡❞✐r ❡ss❛ r❡s♣♦st❛✳ ▼❛s ♦ q✉❡ ❝❤❛♠❛ ♠❛✐s ❛t❡♥çã♦ é q✉❡ ❡❧❡s ✉s❛r❛♠ ❛ ❢ór♠✉❧❛ r❡s♦❧✉t✐✈❛ ❞❡ ❇❤❛s❦❛r❛ s❡♠ ♦ ♣♦❧✐♥ô♠✐♦ s❡r ❞❡ ❣r❛✉ ✷✱ ❝♦♠♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ❛❜❛✐①♦✳

✶✳✶✳✺ ◗✉❡stã♦ ✺

❉❡s❡♥✈♦❧✈❡♥❞♦ ♦ ♣♦❧✐♥ô♠✐♦Q(x) = (2x₋3)2_{✱ ♦❜t❡♠♦s✿}

✭❛✮ 4x2

−12x₋9 ✭❝✮ 4x2

−12x+ 9

✭❜✮ 4x2

+ 12x+ 9 ✭❞✮ 4x2

+ 12x₋9

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈

(31)

✶✳✶✳ ❆♥á❧✐s❡ ❞❛s q✉❡stõ❡s ✷✾

❚❛❜❡❧❛ ✺ ✕ ◗✉❡stã♦ ✺❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✸✱✹

❜ ✷✼✱✻

❝ ✺✶✱✼

❞ ✶✵✱✸

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✼✱✵

❊ss❛ q✉❡stã♦ ♣♦❞❡r✐❛ r❡s♦❧✈✐❞❛ ❞❡ ❞✉❛s ❢♦r♠❛s✿ ❛ ♣r✐♠❡✐r❛ ❡❢❡t✉❛♥❞♦ ❛ ♠✉❧t✐♣❧✐✲ ❝❛çã♦ (2x₋3)_· (2x₋ 3) ❡ ❞❡♣♦✐s ❛❣r✉♣❛♥❞♦ ♦s t❡r♠♦s s❡♠❡❧❤❛♥t❡s ❡ ❛ s❡❣✉♥❞❛ ♣❡❧❛

❢ór♠✉❧❛ ❞♦ q✉❛❞r❛❞♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s t❡r♠♦s✳ ❆ ♣r✐♠❡✐r❛ ❢♦r♠❛ ❢♦✐ ❡s❝♦❧❤✐❞❛ ♣♦r ✶✵✱✸✪ ❞♦s ❛❧✉♥♦s ❡ t♦❞♦s ❡❧❡s ✜③❡r❛♠ ♦ ❝á❧❝✉❧♦ ❝♦rr❡t❛♠❡♥t❡✳ ❆ s❡❣✉♥❞❛ ❢♦r♠❛ ❢♦✐ ❡s✲ ❝♦❧❤✐❞❛ ♣♦r ✶✸✱✽✪ ❞♦s ❞✐s❝❡♥t❡s✱ ♠❛s s♦♠❡♥t❡ ✺✵✪ ❞❡ss❡s ✜③❡r❛♠ ♦ ❝á❧❝✉❧♦ ❝♦rr❡t♦✳ ❖s ♦✉tr♦s ✼✺✱✾✪ ♥ã♦ ✜③❡r❛♠ ❝á❧❝✉❧♦ ❛❧❣✉♠ ❡ ❞❡ss❡s ✶✼✱✷✪ ❛ss✐♥❛❧❛r❛♠ ❝♦rr❡t❛♠❡♥t❡ ❛ q✉❡s✲ tã♦✱ ♦ q✉❡ ❧❡✈❛✲s❡ ❛♦ s❡❣✉✐♥t❡ q✉❡st✐♦♥❛♠❡♥t♦✿ ❊❧❡s r❡❛❧♠❡♥t❡ s❛❜✐❛♠ r❡s♦❧✈❡r ❛ q✉❡stã♦❄✳ ❊♠ ✉♠ ❞♦s ❝á❧❝✉❧♦s ❢❡✐t♦s ♣❡❧♦s ❞✐s❝❡♥t❡s ♦❜s❡r✈❛✲s❡ ❛ ❜✉s❝❛ ✐♥❝❛♥sá✈❡❧ ♣❡❧❛s r❛í③❡s ❞❡ ❡q✉❛çõ❡s✱ ♠❡s♠♦ s❡♠ s❡r ♣❡❞✐❞♦ ♥❛ q✉❡stã♦✱ ♦ ♠❛✐s ✉♠❛ ✈❡③ ✈❡♠ ♠♦str❛r ❛ ❞✐✜❝✉❧❞❛❞❡s ❞❡ ✐♥t❡r♣r❡t❛r ✉♠ t❡st♦ s✐♠♣❧❡s✳ ❖ q✉❡ ✜❝❛ ❝❧❛r♦ ❝♦♠ ❡ss❛ ❛♥á❧✐s❡ é q✉❡ ♠❡s♠♦ ❛ q✉❡stã♦ s❡♥❞♦ ❞❡ ♥í✈❡❧ ❢á❝✐❧ ✹✽✱✸✪ ❞♦s ❛❧✉♥♦s q✉❡ ✜③❡r❛♠ ♦ t❡st❡ ♥ã♦ ❞✐s♣õ❡♠ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦❜t❡r ê①✐t♦ ♥❛ r❡s♦❧✉çã♦ ❞❡ss❛ ❛t✐✈✐❞❛❞❡✳

✶✳✶✳✻ ◗✉❡stã♦ ✻

❆ ❡①♣r❡ssã♦ V(x) =x3 r❡♣r❡s❡♥t❛ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ ❡

x❛ ♠❡❞✐❞❛ ❞❛ ❛r❡st❛ ❞❡ ❝❛❞❛ ❢❛❝❡✳ ❙❡V(x) = 343 cm3✱ ❡♥tã♦ ♦ ✈❛❧♦r ❞❡

x é✿

✭❛✮ 70 _✭❜✮

71 _✭❝✮

72 _✭❞✮

73

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❇

❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✻✳

❚❛❜❡❧❛ ✻ ✕ ◗✉❡stã♦ ✻❚✶ ❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✵

❜ ✶✼✱✷

❝ ✸✱✹

❞ ✼✾✱✹

(32)

✸✵ ❈❛♣ít✉❧♦ ✶✳ ❚❡st❡ ✶ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

❛t❡♥çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ♣❡r❝❡❜❡r q✉❡ ♦ ✈❛❧♦r ❞❛ ♠❡❞✐❞❛ ❞❛ ❛r❡st❛ ❡r❛ ❛ ❜❛s❡ ❞❛ ♣♦tê♥❝✐❛ ❡ ♥ã♦ ❛ ♣ró♣r✐❛ ♣♦tê♥❝✐❛✱ ♠❡s♠♦ s❡♥❞♦ ❞✐t♦ ♥♦ ❡♥✉♥❝✐❛❞♦ ❞❛ q✉❡stã♦ ❡ ❡❧❡s ❥á t❡r❡♠ ❡st✉❞❛❞♦ ❝♦♠♦ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❡①♣♦♥❡♥❝✐❛✐s✳ ▼❛s ✈❛❧❡ r❡ss❛❧t❛r q✉❡ ❡❧❡s s♦✉❜❡r❛♠ ❞❡❝♦♠♣♦r ✸✹✸ ❡♠ ❢❛t♦r❡s ♣r✐♠♦s✳

✶✳✶✳✼ ◗✉❡stã♦ ✼

P❛r❛ q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ P(h) = h2

−Ah+ 16s❡❥❛ ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✱ ♦ ✈❛❧♦r ❞❡A ❞❡✈❡ s❡r✿

✭❛✮ 2 + 2 ✭❜✮ 2 + 4 ✭❝✮ 2 + 6 ✭❞✮ 2 + 8

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈

❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛♠ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭❞❡ ✉♠ t♦t❛❧ ❞❡ ✷✾ ♣❛rt✐❝✐♣❛♥t❡s✮ q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✼✳

❚❛❜❡❧❛ ✼ ✕ ◗✉❡stã♦ ✼❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✹✶✱✹

❜ ✷✵✱✼

❝ ✷✵✱✼

❞ ✶✸✱✽

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✸✱✹

❊ss❛ q✉❡stã♦ é ❞❡ ♥í✈❡❧ ❜ás✐❝♦ ❡ ♠❡s♠♦ ❛ss✐♠ s♦♠❡♥t❡ ✷✵✱✼✪ ❞♦s ❛❧✉♥♦s ♠❛r❝❛✲ r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❈✱ q✉❡ é ❛ ❛❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✳ ❏á ✹✶✱✹✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❆✱ ❡sq✉❡❝❡♥❞♦ q✉❡ ❞❡✈❡r✐❛ ♠✉❧t✐♣❧✐❝❛r ♣♦r ✷ ♣❛r❛ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ❝♦rr❡t♦✳ ▼❛s ♦❜s❡r✈❛✲s❡ ❛✐♥❞❛ q✉❡ ❛s ❛❧t❡r♥❛t✐✈❛s ❇ ❡ ❈ t✐✈❡r❛♠ ❛ ♠❡s♠❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ❡s❝♦❧❤❛✱ ♦ q✉❡ ✈❡♠ ♠❛✐s ❛ ❡✈✐❞❡♥❝✐❛r q✉❡ ♦s ❛❧✉♥♦s ♥ã♦ ❞♦♠✐♥❛♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣r♦❞✉t♦s ♥♦tá✈❡✐s✳

✶✳✶✳✽ ◗✉❡stã♦ ✽

❆ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ é ❞❛❞❛ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ A(x) = 4x2

+ 24x+ 36✱ ❡♥tã♦ ♦ ❜✐♥ô♠✐♦

q✉❡ r❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r ❞♦ ❧❛❞♦ ❞❡ss❡ q✉❛❞r❛❞♦ é✿

✭❛✮ 2(x+ 3) ✭❜✮ 2(x₋3) ✭❝✮ 4(x₋3) ✭❞✮ 4(x+ 3)

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❆

(33)

✶✳✶✳ ❆♥á❧✐s❡ ❞❛s q✉❡stõ❡s ✸✶

❚❛❜❡❧❛ ✽ ✕ ◗✉❡stã♦ ✽❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✹✶✱✹

❜ ✶✵✱✸

❝ ✶✼✱✷

❞ ✷✹✱✶

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✼✱✵

P❛r❛ r❡s♦❧✈❡r ❡ss❛ q✉❡stã♦ ♦ ❞✐s❝❡♥t❡ ❞❡✈❡r✐❛ ❝♦❧♦❝❛r ♦ ✹ ❡♠ ❡✈✐❞ê♥❝✐❛✱ ♦✉ s❡❥❛✱

4(x2

+ 6x+ 9)✱ ❡♠ s❡❣✉✐❞❛ ❡s❝r❡✈❡r ♦ tr✐♥ô♠✐♦(x2

+ 6x+ 9) = (x+ 3)2 _{❝♦♠♦ ✉♠ ♣r♦❞✉t♦}

♥♦tá✈❡❧ ❡ ❞❡♣♦✐s ❡①tr❛✐r ❛ r❛✐③ q✉❛❞r❛❞❛ ❞♦ ♣r♦❞✉t♦4(x+3)2_{✳ ❆ ❛❧t❡r♥❛t✐✈❛ ❆ ✭❝♦rr❡t❛✮ ❢♦✐}

❡s❝♦❧❤✐❞❛ ♣♦r ✹✶✱✹✪ ❞♦s ❛❧✉♥♦s ❡ ✷✹✱✶✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❉✱ ♦✉ s❡❥❛✱ ❛❧❣✉♥s ❛❧✉♥♦s ❝♦♥❢✉♥❞✐r❛♠ ❛s r❡s♣♦st❛s ♣♦rq✉❡ ♥ã♦ ❡①tr❛ír❛♠ ❛ r❛✐③ q✉❛❞r❛❞❛ ❡ ✶✵✱✸✪ ❞♦s ❞✐s❝❡♥t❡s ♥ã♦ s♦✉❜❡r❛♠ ❞✐❢❡r❡♥❝✐❛r ♦ q✉❛❞r❛❞♦ ❞❛ s♦♠❛ ❡ ♦ q✉❛❞r❛❞♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s t❡r♠♦s✱ s❡♥❞♦ ♠❛✐s ✉♠❛ ✈❡③ ❝♦♥✜r♠❛❞❛ ❛ ❞✐✜❝✉❧❞❛❞❡ q✉❡ ❡❧❡s t❡♠ ♥❡ss❡ ❛ss✉♥t♦✳

✶✳✶✳✾ ◗✉❡stã♦ ✾

❙❡❥❛♠ A(x) ❡ B(x) ♦s ♣♦❧✐♥ô♠✐♦s q✉❡ r❡♣r❡s❡♥t❛♠ ❛s ár❡❛s ❞❛s ✜❣✉r❛s ♥❛s ❝♦r❡s r♦s❛ ❡

❧✐❧ás✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ♣♦❧✐♥ô♠✐♦ P(x) = A(x) +B(x) é ❞❛❞♦ ♣♦r✿

x+ 4

A(x)

B(x) x

x+ 2

✭❛✮ 2x2

+ 8x+ 8 ✭❝✮ 2x2

+ 8x+ 4

✭❜✮ 2x2

+ 4x+ 4 ✭❞✮ x2

+ 4x

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈

(34)

✸✷ ❈❛♣ít✉❧♦ ✶✳ ❚❡st❡ ✶ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

❚❛❜❡❧❛ ✾ ✕ ◗✉❡stã♦ ✾❚✶ ❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✶✼✱✷

❜ ✸✶✱✵

❝ ✹✹✱✽

❞ ✼✱✵

✹✹✱✽✪ ❞♦s ❛❧✉♥♦s ❛❝❡rt❛r❛♠ ❡ss❛ q✉❡stã♦ ✭❛❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ❈✮✱ ✐ss♦ ✐♥❞✐❝❛ q✉❡ ❡❧❡s t❡♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❝♦♠♦ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ r❡tâ♥❣✉❧♦ ✭❞❡❝♦r❛♠ ❢ór♠✉❧❛s✮✱ ♠❛s s♦♠❡♥t❡ ❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♥ã♦ ❡r❛ s✉✜❝✐❡♥t❡ ♣❛r❛ s♦❧✉❝✐♦♥❛r ❛ q✉❡stã♦ ✾✳ ❉♦s ❛❧✉♥♦s q✉❡ ❛❝❡rt❛r❛♠ ❡ss❛ q✉❡stã♦ s♦♠❡♥t❡ ✶✸✱✽✪ ✜③❡r❛♠ ♦s ❝á❧❝✉❧♦ ♥❡❝❡ssár✐♦s ❡ t♦❞♦s ❝♦rr❡t❛♠❡♥t❡✱ ♠❛s ✷✵✱✼✪ só ❛ss✐♥❛❧❛r❛♠ s❡♠ ❞❡♠♦str❛r ♦ ♣♦rq✉❡ ❡s❝♦❧❤❡r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❆✳ ❖ q✉❡ ❧❡✈❛ ❛ q✉❡st✐♦♥❛r s❡ ❡❧❡s s❛❜✐❛♠ ♠❡s♠♦ ❝♦♠♦ r❡s♦❧✈❡r ❛ q✉❡stã♦ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❛ss✐♥❛❧❛r❛♠ q✉❛❧q✉❡r ❛❧t❡r♥❛t✐✈❛ ✈✐st♦ q✉❡ ❡❧❡s ❡rr❛r❛♠ q✉❡stõ❡s ♠❛✐s s✐♠♣❧❡s ❞♦ q✉❡ ❡ss❛✳

✶✳✶✳✶✵ ◗✉❡stã♦ ✶✵

❈♦♥s✐❞❡r❡ P(x) = x2

−25 ❡ D(x) =x₋5✳ ❖ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ ❞❡ P(x) ♣♦r D(x) é

✐❣✉❛❧ ❛✿

✭❛✮ x₋5 ✭❜✮ x₋25 ✭❝✮ x+ 5 ✭❞✮ x+ 25

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈

P♦r ✐♥t❡r♠é❞✐♦ ❞❛ t❛❜❡❧❛ ❛❜❛✐①♦ ♣♦❞❡✲s❡ ✈❡r ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✶✵✳

❚❛❜❡❧❛ ✶✵ ✕ ◗✉❡stã♦ ✶✵❚✶ ❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✸✼✱✾

❜ ✶✸✱✽

❝ ✸✹✱✺

❞ ✶✸✱✽

(35)

✶✳✶✳ ❆♥á❧✐s❡ ❞❛s q✉❡stõ❡s ✸✸

✶✳✶✳✶✶ ◗✉❡stã♦ ✶✶

❙❛❜❡♥❞♦ q✉❡R(a)₋F(a) = 0 ❡ q✉❡F(a) = 6a3

+ 5a2

−3a₋2✱❡♥tã♦ ♦ ♣♦❧✐♥ô♠✐♦R(a)é✿

✭❛✮ −6a3

−5a2

+ 3a+ 2 ✭❝✮ 6a3

+ 5a2

−3a₋2

✭❜✮ 6a3

+ 5a2

−3a+ 2 ✭❞✮ 6a3

+ 5a2

+a+ 2

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈

❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✶✶✳

❚❛❜❡❧❛ ✶✶ ✕ ◗✉❡stã♦ ✶✶❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✺✽✱✼

❜ ✶✼✱✷

❝ ✷✵✱✼

❞ ✵

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✸✱✹

▼❡s♠♦ ❛ q✉❡stã♦ s❡♥❞♦ ❞❡ ❡①tr❡♠❛♠❡♥t❡ ❢á❝✐❧ ♣❛r❛ ❛ sér✐❡ ❝✉rs❛❞❛ ♣❡❧♦s ❛❧✉♥♦s s♦♠❡♥t❡ ✷✵✱✼✪ ❛❝❡rt❛r❛♠ ✭❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❈✮ ❡ ❛✐♥❞❛ ❛ss✐♠ ✼✾✱✸✪ ❞♦s ♠❡s♠♦s ❡rr❛r❛♠ ❛ q✉❡stã♦✳ ❖❜s❡r✈❛♥❞♦ ❛ ❚❛❜❡❧❛ ✶✶ ✈❡r✐✜❝❛✲s❡ q✉❡ ✺✽✱✼✪ ❞♦s ❞✐s❝❡♥t❡s ❛ss✐♥❛✲ ❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❆✱ ♦ q✉❡ ❞❡✐①❛ ❡✈✐❞❡♥t❡ q✉❡ ❡ss❡s ❛❧✉♥♦s t❡♠ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s✱ ♠❡s♠♦ s❡♥❞♦ ❛ ♠❛✐s s✐♠♣❧❡s ♣♦ssí✈❡❧✳

✶✳✶✳✶✷ ◗✉❡stã♦ ✶✷

❊❢❡t✉❛♥❞♦ ❛ ❞✐✈✐sã♦ ❞❡2x2

+ 5x+ 3 ♣♦rx+ 2✱ ♦❜t❡♠♦s ♦ r❡st♦ R(x)❡ ♦ q✉♦❝✐❡♥t❡ Q(x)✱

❝♦♠♦ ✈❡♠♦s ❛❜❛✐①♦✿

2x2

+ 5x+ 3 _| x+ 2

R(x) Q(x)

❆ ❛❧t❡r♥❛t✐✈❛ q✉❡ ❝♦♥té♠R(x)é✿

✭❛✮ 0 ✭❜✮ 1 ✭❝✮ 2 ✭❞✮ 3

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❇

(36)

✸✹ ❈❛♣ít✉❧♦ ✶✳ ❚❡st❡ ✶ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

❚❛❜❡❧❛ ✶✷ ✕ ◗✉❡stã♦ ✶✷❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✵

❜ ✷✼✱✻

❝ ✹✹✱✽

❞ ✷✵✱✼

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✻✱✾

❖❜s❡r✈❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❛ ❚❛❜❡❧❛ ✶✷✱ ♣❡r❝❡❜❡✲s❡ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ✭✹✹✱✽✪✮ ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❈ ❡ ✷✼✱✻✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❇✱ q✉❡ é ❛ ❝♦rr❡t❛✳ ❖❜s❡r✈❛♥❞♦ ♦s ❞❛❞♦s ❛❝✐♠❛ ✈❡r✐✜❝❛✲s❡ q✉❡ ♠❡s♠♦ ♦s ❛❧✉♥♦s t❡♥❞♦ ✈✐st♦ ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ♥♦ ✽➸ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ♥ã♦ ❢♦✐ ❡✜❝❛③ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ss❡ ❛ss✉♥t♦✱ ♦ q✉❡ ✐rá ❞✐✜❝✉❧t❛r ❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ♦✉tr♦s ❝♦♥t❡ú❞♦s q✉❡ ♦ t❡♥❤❛♠ ❝♦♠♦ ♣ré✲r❡q✉✐s✐t♦✳

✶✳✶✳✶✸ ◗✉❡stã♦ ✶✸

❆s r❛í③❡s ❞♦ ♣♦❧✐♥ô♠✐♦ M(x) = x2

− 3₂x+ 1

2✱ ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡✱ sã♦✿

✭❛✮ 1 ❡ ₋1

2 ✭❜✮ 1 ❡

1

2 ✭❝✮ −

1

2 ❡1 ✭❞✮

1 2 ❡ 1

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❉

❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭❞❡ ✉♠ t♦t❛❧ ❞❡ ✷✾ ♣❛rt✐❝✐♣❛♥t❡s✮ q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✶✸✳

❚❛❜❡❧❛ ✶✸ ✕ ◗✉❡stã♦ ✶✸❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✶✵✱✸

❜ ✸✹✱✺

❝ ✷✼✱✻

❞ ✷✵✱✼

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✻✱✾

(37)

✶✳✶✳ ❆♥á❧✐s❡ ❞❛s q✉❡stõ❡s ✸✺

✶✳✶✳✶✹ ◗✉❡stã♦ ✶✹

❋❛t♦r❛♥❞♦ ♦ ♣♦❧✐♥ô♠✐♦ Q(y) =y4

−5y2

+ 4✱ ♦❜t❡♠♦s✿

✭❛✮ Q(x) = (y₋1)(y+ 1)(y₋2)(y+ 2)

✭❜✮ Q(x) = (y+ 1)(y+ 1)(y₋2)(y+ 2)

✭❝✮ Q(x) = (y₋1)(y+ 1)(y+ 2)(y+ 2)

✭❞✮ Q(x) = (y₋1)(y₋1)(y₋2)(y+ 2)

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❆

❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭❞❡ ✉♠ t♦t❛❧ ❞❡ ✷✾ ♣❛rt✐❝✐♣❛♥t❡s✮ q✉❡ ❛ss✐♥❛❧❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ q✉❡stã♦ ✶✹✳

❚❛❜❡❧❛ ✶✹ ✕ ◗✉❡stã♦ ✶✹❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✷✵✱✼

❜ ✶✼✱✷

❝ ✸✽✱✵

❞ ✶✸✱✽

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✶✵✱✸

✸✽✱✵✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❈✱ ♠❛s s♦♠❡♥t❡ ✷✵✱✼✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❝♦rr❡t❛ ✭❛❧✲ t❡r♥❛t✐✈❛ ❆✮✳ ❖❜s❡r✈❛✲s❡ q✉❡ ✶✵✱✸✪ ♥ã♦ ❛ss✐♥❛❧♦✉ ♥❡♥❤✉♠❛ ❞❛s ❛❧t❡r♥❛t✐✈❛s ❡ ♥♦✈❛♠❡♥t❡ ✜❝❛ ❛ ✐♥❝ó❣♥✐t❛ ❞♦ ♣♦r q✉ê✳ ❊ss❛ q✉❡stã♦ ♣♦❞❡r✐❛ s❡r r❡s♦❧✈✐❞❛ ✉t✐❧✐③❛♥❞♦ ❛s ❡q✉❛çõ❡s ❜✐q✉❛❞r❛❞❛s ✈✐st❛s ♥♦ ✾➸ ❛♥♦ ❡ ♠❛✐s ✉♠❛ ✈❡③ ❝♦♥st❛t❛✲s❡ ❛ ❞❡✜❝✐ê♥❝✐❛ ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ á❧❣❡❜r❛ ❞♦s ♣♦❧✐♥ô♠✐♦s✳

✶✳✶✳✶✺ ◗✉❡stã♦ ✶✺

❆ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ x2

−x + 3 = 0 ♣❡rt❡♥❝❡♠ ❛♦

❝♦♥❥✉♥t♦✿

✭❛✮ {−1,1,2_} ✭❜✮ _{−1,1,3_} ✭❝✮ _{−2,1,2_} ✭❞✮ _{−1,2,3_}

❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❇

(38)

✸✻ ❈❛♣ít✉❧♦ ✶✳ ❚❡st❡ ✶ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

❚❛❜❡❧❛ ✶✺ ✕ ◗✉❡stã♦ ✶✺❚✶

❆❧t❡r♥❛t✐✈❛ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s

❛ ✷✹✱✶

❜ ✸✹✱✺

❝ ✷✹✱✶

❞ ✶✵✱✸

♥✳❛✳ ✭♥ã♦ ❛ss✐♥❛❧♦✉✮ ✼✱✵

❊ss❛ ú❧t✐♠❛ q✉❡stã♦ ✈✐s❛✈❛ ✈❡r✐✜❝❛r s❡ ♦ ❛❧✉♥♦ s❛❜✐❛ ✐❞❡♥t✐✜❝❛r ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❣r❛✉ ✷ s❡♠ ❡❢❡t✉❛r ♦ ❝á❧❝✉❧♦ ❞❛s r❛í③❡s ♣❡❧❛ ❢ór♠✉❧❛ ✉s✉❛❧✱ ♣♦ré♠ ✻✺✱✺✪ ❛✐♥❞❛ ❡rr❛r❛♠ ❡ss❛ q✉❡stã♦ q✉❡ é ❞❡ ♥í✈❡❧ ♠✉✐t♦ ❢á❝✐❧ ♣❛r❛ ❛ sér✐❡ q✉❡ ❡❧❡s ❡stã♦ ❝✉rs❛♥❞♦✳ ❙♦♠❡♥t❡ ✸✹✱✺✪ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✭❇✮✳ ❖ q✉❡ ♠❛✐s ✉♠❛ ✈❡③ ❝♦♠♣r♦✈❛ ❛ ❞❡✜❝✐ê♥❝✐❛ ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ tr❛❜❛❧❤❛r ❡ss❡ ❝♦♥t❡ú❞♦ ❞❡ ❢♦r♠❛ ♠✐♥✉❝✐♦s❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

✶✳✷ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s

❖ ❣rá✜❝♦ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭❞❡ ✉♠ t♦t❛❧ ❞❡ ✷✾ ♣❛rt✐❝✐♣❛♥t❡s✮ q✉❡ ❛❝❡rt❛r❛♠ ❝❛❞❛ q✉❡stã♦ ❞♦ ❚❡st❡ ✶✳ ❖ ♣❡r❝❡♥t✉❛❧ ♠é❞✐♦ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦ ❢♦✐ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✸✵✱✶✪✳

(31%) (34.5%) (20.7%) (31%)

(51.7%)

(17.2%) (20.7%)

(41.4%)

(44.8%)

(34.5%)

(20.7%) (27.6%) (20.7%)

(20.7%) (34.5%)

Percentual de acertos por questão

(39)

✶✳✷✳ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✸✼

❆♣ós ❛♥❛❧✐s❛r ♦ ❣rá✜❝♦ ❛❝✐♠❛✱ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡ ❡♠ ✾✸✱✸✪ ❞❛s q✉❡stõ❡s ♦ í♥❞✐❝❡ ❞❡ ❛❝❡rt♦s ♥ã♦ ❢♦✐ s❛t✐s❢❛tór✐♦✳ ◆♦ ❣rá✜❝♦ ✈❡r✐✜❝❛✲s❡ ❛✐♥❞❛ q✉❡ ❛ q✉❡stã♦ ❝♦♠♦ ♦ ♠❛✐♦r í♥❞✐❝❡ ❞❡ ❛❝❡rt♦ ❢♦✐ ❛ ✺ ❝♦♠ ✺✶✱✼✪ ❡ ❛ q✉❡stã♦ ❝♦♠ ♦ ♠❡♥♦r í♥❞✐❝❡ ❞❡ ❛❝❡rt♦ ❢♦✐ ❛ q✉❡stã♦ ✻ ❝♦♠ ✶✼✱✷✪✱ ♠❡s♠♦ s❡♥❞♦ ✉♠❛ q✉❡stã♦ ❜❛st❛♥t❡ s✐♠♣❧❡s✳ ❈♦♠ ❜❛s❡ ♥❛ ❛♥á❧✐s❡ ❝♦♥st❛t❛✲s❡ q✉❡ ♦s ❞✐s❝❡♥t❡s ♥ã♦ ❡♥t❡♥❞❡r❛♠ ❛ á❧❣❡❜r❛ ❞♦s ♣♦❧✐♥ô♠✐♦s ❡♥s✐♥❛❞❛ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♠❡s♠♦ ❡❧❡s t❡♥❞♦ s✐❞♦ ❛♣r♦✈❛❞♦s ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ♠❛t❡♠át✐❝❛✱ ❡ ♣♦r ❡ss❡ ♠♦t✐✈♦ ♠♦str❛♠ ❣r❛♥❞❡s ❞✐✜❝✉❧❞❛❞❡s ❡♠ ✉t✐❧✐③❛r ♦ ♣♦✉❝♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ❢♦✐ ✏❛❞q✉✐r✐❞♦✑✳ ❊♠ r❡s✉♠♦✱ ♦s ❛❧✉♥♦s ♥ã♦ ❝♦♥s❡❣✉✐r❛♠ t❡r ❡❢❡t✐✈♦ ❛❝❡ss♦ ❛ ❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❝✉❥❛ ✐♠♣♦rtâ♥❝✐❛✱ ♥ã♦ só ♣❛r❛ ♦ ♠❡✐♦ ❛❝❛❞ê♠✐❝♦✱ ♠❛s t❛♠❜é♠ ♣❛r❛ ❛ ✈✐❞❛ é ✐♠♣r❡s❝✐♥❞í✈❡❧✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ▼✐r❛♥❞❛ ✭❛♣✉❞ ●❘❆◆❉❖✱ ✷✵✵✻✮✿

[. . .]♦s ❝♦♥❝❡✐t♦s ❞♦ ❝❛♠♣♦ ❛❧❣é❜r✐❝♦ ❝♦♥st✐t✉❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥❤❡✲

❝✐♠❡♥t♦s ❜❛st❛♥t❡ s✐❣♥✐✜❝❛t✐✈♦s ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❞❡s❡♥✈♦❧✈❛ s✉❛ ❝❛♣❛✲ ❝✐❞❛❞❡ ❞❡ ❛♥á❧✐s❡ ❡ sí♥t❡s❡✱ ❞❡ ❛❜str❛çã♦ ❡ ❣❡♥❡r❛❧✐③❛çã♦✱ ❛❧é♠ ❞❡ ❧❤❡ ♣♦ss✐❜✐❧✐t❛r ❛ ❛q✉✐s✐çã♦ ❞❡ ✉♠❛ ♣♦❞❡r♦s❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ✭♣✳ ✺✻✮✳

(40)

(41)

✸✾

✷ ❖ ❡st✉❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦♠ r❡❧❛t♦s ❞❡ ❤✐s✲

tór✐❛ ❞❛ ♠❛t❡♠át✐❝❛

◆❡st❡ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛❞♦ ♦ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ q✉❡ ❢♦✐ ❡❧❛❜♦r❛❞♦ ♣❛r❛ s❡r ✉s❛❞♦ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠ ❛ t✉r♠❛ q✉❡ ❢❡③ ♦ ❚❡st❡ ✶✳ ❊ss❡ ♠❛t❡r✐❛❧ ❡①♣❧♦r❛ ♦ ❡st✉❞♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ❡ ✉t✐❧✐③❛ tó♣✐❝♦s ❞❡ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ♣❛r❛ ❞❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ❞♦s ❞✐s❝❡♥t❡s ♣❡❧♦ ❝♦♥t❡ú❞♦ ♠✐♥✐str❛❞♦✱ t❛♠❜é♠ tr❛③ ✉♠ t❡①t♦ ❝♦♠ ✉♠❛ ❡stét✐❝❛ ♠❛✐s ❛tr❛t✐✈❛ ✈✐s✉❛❧♠❡♥t❡✳ ❖s ♣❧❛♥♦s ❞❛s ❛✉❧❛s ♠✐♥✐str❛❞❛s ♣❛r❛ ♦s ❞✐s❝❡♥t❡s s❡ ❡♥❝♦♥tr❛♠ ♥♦s ❛♥❡①♦s✳ ❖ ■❋❈❊ ♦❢❡r❡❝❡ ✻✵ ✈❛❣❛s ♣❛r❛ ❝✉rs♦ ❚é❝♥✐❝♦ ❡♠ ■♥❢♦r♠át✐❝❛ ■♥t❡❣r❛❞♦ ❛♦ ❊♥s✐♥♦ ▼é❞✐♦ t♦❞♦ ❛♥♦✱ ♣♦ré♠ ❡♠ ✷✵✶✺ ❛s ✈❛❣❛s ♥ã♦ ❢♦r❛♠ ♣r❡❡♥❝❤✐❞❛s ❡ s♦♠❡♥t❡ ✹✽ ❛❧✉♥♦s ❢♦r❛♠ ♠❛tr✐❝✉❧❛❞♦s✳ ❋♦r❛♠ ❝r✐❛❞❛s ❞✉❛s t✉r♠❛s ❝❛❞❛ ✉♠❛ ❝♦♠ ✷✹ ❛❧✉♥♦s✱ ♣♦ré♠ ❤♦✉✈❡ ✻ tr❛♥s❢❡rê♥❝✐❛s ❧♦❣♦ ♥♦ ♣r✐♠❡✐r♦ ❜✐♠❡str❡ ❡ ✽ ❞❡s✐stê♥❝✐❛s ♥♦ ❞❡❝♦rr❡r ❞♦s ♦✉tr♦s ❞♦✐s ❜✐♠❡str❡s✳ ❊♥tã♦✱ ❛ ❝♦♦r❞❡♥❛çã♦ ❞❡❝✐❞✐✉ ❥✉♥t❛r ❛s ❞✉❛s t✉r♠❛s ❡ ❤♦❥❡ só ❤á ✉♠❛ t✉r♠❛ ❞❡ ✶➟ sér✐❡ ❞❡ss❡ ❝✉rs♦ té❝♥✐❝♦ ❝♦♠ ✸✹ ❛❧✉♥♦s✱ s❡♥❞♦ q✉❡ ❛❧❣✉♥s ♥ã♦ ❝♦♠♣❛r❡❝❡♠ ❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛ ♥❡ss❡ ✹➸ ❜✐♠❡str❡ ♣♦rq✉❡ ❥á ❡stã♦ ❡♠ ❞❡♣❡♥❞ê♥❝✐❛ ♣❛r❛ ♦ ♣ró①✐♠♦ ❛♥♦✳

❉❡✈✐❞♦ ❛♦s r❡sq✉í❝✐♦s ❞❡ ❞✉❛s ❣r❡✈❡s ♣❡❧❛s q✉❛✐s ♣❛ss❛♠♦s ♦ ❛♥♦ ❧❡t✐✈♦ ❞❡ ✷✵✶✺ só ✐♥✐❝✐♦✉ ❡♠ ♠❛rç♦ ❡ s❡✉ tér♠✐♥♦ ❢♦✐ ❡♠ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻✳ ❋✐❝♦✉ ❛❝♦r❞❛❞♦ ❝♦♠ ❛ ❝♦♦r❞❡♥❛çã♦ q✉❡ s❡r✐❛ ❞❡st✐♥❛❞♦ ❛ ♦✜❝✐♥❛ ♣❛r❛ ❡st✉❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✶✷ ❛✉❧❛s ✭s❡♥❞♦ q✉❡ ✷ ❞❡ss❛s ❛✉❧❛s ❛♣❧✐❝❛çã♦ ❞♦ ❚❡st❡ ✷✮✱ ❝♦♥❢♦r♠❡ t❛❜❡❧❛ ❛❜❛✐①♦✿

❚❛❜❡❧❛ ✶✻ ✕ ❍♦rár✐♦ ❞❡ ❛✉❧❛s ❞❛ ♦✜❝✐♥❛

❉❛t❛ P❧❛♥♦ ❍♦rár✐♦ ❚♦t❛❧ ❞❡ ❛✉❧❛s

✶✶✴✵✶✴✷✵✶✻ ✶ ✶✸✿✶✺ às ✶✹✿✵✺ ✶

✶✺✴✵✶✴✷✵✶✻ ✷ ✼✿✶✺ às ✽✿✺✺ ❡ ✾✿✵✺ às ✾✿✺✺ ✸

✶✽✴✵✶✴✷✵✶✻ ✸ ✶✸✿✶✺ às ✶✹✿✵✺ ✶

✷✷✴✵✶✴✷✵✶✻ ✹ ✼✿✶✺ às ✽✿✺✺ ❡ ✾✿✵✺ às ✾✿✺✺ ✸

✷✻✴✵✶✴✷✵✶✻ ✺ ✶✸✿✶✺ às ✶✹✿✺✺ ✷

✷✾✴✵✶✴✷✵✶✻ ❆♣❧✐❝❛çã♦ ❞♦ ❚❡st❡ ✷ ✼✿✶✺ às ✽✿✺✺ ✷

❖ t❡①t♦ ❛ s❡❣✉✐r ❢♦✐ ❡❧❛❜♦r❛❞♦ ❝♦♠ ❜❛s❡ ♥♦s ❧✐✈r♦s ❡ r❡✈✐st❛ ❝✐t❛❞♦s ❞❡ ❬✶❪ ❛ ❬✶✶❪ ♥❛s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳

✷✳✶ ❖r✐❣❡♠ ❞❛ ➪❧❣❡❜r❛

E

_{r❛ ✉♠❛ ✈❡③✳✳✳}

(42)

✹✵ ❈❛♣ít✉❧♦ ✷✳ ❖ ❡st✉❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦♠ r❡❧❛t♦s ❞❡ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛

✈❛♠♦s ❞❛r ✉♠ ♣❛ss❡✐♦ ❡♠ ✉♠ ♠✉♥❞♦ ❝❤❡✐♦ ❞❡ ❛✈❡♥t✉r❛s ❡ ❢❡✐t♦s✳ ❙❛❜❡ q✉❡ ♠✉♥❞♦ é ❡ss❡❄ ◆ã♦❄ P❡♥s❡ ❜❡♠✳ ➱ ♦ ♥♦ss♦✳ ❊♥tã♦ ❝♦♠❡❝❡♠♦s✳

✶

❋✐❣✉r❛ ✶ ✕ ❍✐♣át✐❛

❆ ❝✐✈✐❧✐③❛çã♦ ❡❣í♣❝✐❛ ❞❡s❡♥✈♦❧✈❡✉✲s❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♥s q✉❛tr♦ ♠✐❧ ❛♥♦s ❡ ❞❡✐①♦✉✲♥♦s ♠❛r❝❛s ♠❛r❛✈✐❧❤♦s❛s✳ ❆s ♠❛✐s ❝♦♥❤❡❝✐❞❛s sã♦✱ ❝❧❛r♦✱ ❛s ♣✐râ♠✐❞❡s ❞❡ ●✐sé ❡ ❛ ❊s✜♥❣❡✳ ❱❛✲ ♠♦s ❛❜♦r❞❛r ✉♠ ♣♦✉❝♦ ❛ ❤❡r❛♥ç❛ ♠❛t❡♠át✐❝❛ ❞❡st❡s ✐❧✉str❡s ❛♥t❡♣❛ss❛❞♦s✳ ❆ ♥♦ss❛ ❢♦♥t❡ ♣r✐♥❝✐♣❛❧ é ✉♠ ♣❛♣✐r♦✱ ❝♦♥✲ t❡♥❞♦ ♣r♦❜❧❡♠❛s ❞❡ ♠❛t❡♠át✐❝❛✱ ❡s❝r✐t♦ ♣♦r ✈♦❧t❛ ❞❡ ✶✻✺✵ ❛✳❊✳❈✳ ❊st❡ ❞♦❝✉♠❡♥t♦ ❝♦♥té♠ ✽✺ ❡♥✉♥❝✐❛❞♦s ❝♦♣✐❛❞♦s ❡♠ ❡s❝r✐t❛ ❤✐❡rát✐❝❛ ✭❡s❝r✐t❛ ❤✐❡r♦❣❧í✜❝❛ s✐♠♣❧✐✜❝❛❞❛ ✉s❛❞❛ ♣❛r❛ ❡s❝r❡✈❡r t❡①t♦s ❝♦♠ ✉♠ ♣✐♥❝❡❧ ❡♠ t✐r❛s ❞❡ ♣❛♣✐r♦✮✱ ❝✉❥♦ ❛✉t♦r ❢♦✐ ❆❤♠❡s✱ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ ♥♦♠❡ ❞♦ ❤✐st♦r✐❛❞♦r ❡s❝♦❝ês q✉❡ ♦ ❝♦♠♣r♦✉ ♥♦ sé❝✉❧♦ ❳■❳✱ ❆❧❡①❛♥❞❡r ❍❡♥r② ❘❤✐♥❞✳ ❖ ♣❛♣✐r♦ ❞❡ ❘❤✐♥❞ é ✉♠❛ ❢♦♥t❡ ♣r✐♠ár✐❛ r✐❝❛ s♦❜r❡ ❛ ♠❛t❡♠át✐❝❛ ❡❣í♣❝✐❛ ❛♥t✐❣❛✱ ❞❡s❝r❡✈❡ ♦s ♠ét♦❞♦s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦✱ ♦ ✉s♦ q✉❡ ❢❛③✐❛ ❞❛s ❢r❛çõ❡s ✉♥✐tár✐❛s✱ s❡✉ ❡♠♣r❡❣♦ ❞❛ r❡❣r❛ ❞❛ ❢❛❧s❛ ♣♦s✐çã♦✱ s✉❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ ❡ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ❞❛ ♠❛t❡♠át✐❝❛ ❛ s✐t✉❛çõ❡s ♣rát✐❝❛s✳✷ _{❱❡❥❛♠♦s ✉♠} ❡①❡♠♣❧♦✿

Problema 24:

❖ ✈❛❧♦r ❞❡ ✏❛❤❛✑ s❡ ✏❛❤❛✑ ❡ ✉♠ sét✐♠♦ ❞❡ ✏❛❤❛✑ é ✶✾✳

P❛r❛ s♦❧✉❝✐♦♥❛r ♦ ♣r♦❜❧❡♠❛ ♦s ❡❣í♣❝✐♦s ✉t✐❧✐③❛✈❛♠ ✉♠❛ té❝♥✐❝❛ ❞❡♥♦♠✐♥❛❞❛

método

falsa posição

✳ ❊ss❡ ♠ét♦❞♦ ❝♦♥s✐st✐❛ ❡♠ ❡s❝♦❧❤❡r ✉♠ ✈❛❧♦r ❛r❜✐trár✐♦ ♣❛r❛ ✏❛❤❛✑ ❡ ❛ ♣❛rt✐r ❞❡st❡ ✈❛❧♦r ❡❧❡s ❢❛③✐❛♠ ♦s ❝á❧❝✉❧♦s ❡ ❝♦♠♣❛r❛✈❛♠ ❝♦♠ ♦ r❡s✉❧t❛❞♦✱ ♠❛s ♣r♦✈❛✈❡❧♠❡♥t❡ ♥ã♦ ❡r❛ ♦ r❡s✉❧t❛❞♦ ❡s♣❡r❛❞♦✳ P♦r ✐ss♦ ❡❧❡s ✉t✐❧✐③❛✈❛♠ ✉♠ ❢❛t♦r ❞❡ ❝♦rr❡çã♦ ♣❛r❛ ♦❜t❡r ♦ ✈❛❧♦r ❝♦rr❡t♦ ❞❡ ✏❛❤❛✑✱ ♦✉ s❡❥❛✱ ♦ ✈❛❧♦r q✉❡ s❛t✐s❢❛③ ❛ ❡①♣r❡ssã♦✳ ❙❡❣✉✐♥❞♦ ♦ ♠ét♦❞♦ ❡❣í♣❝✐♦ ✈❛♠♦s r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❡ ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❡ ✏❛❤❛✑✳

Solução:

❙❡❥❛ ✏❛❤❛✑= 7✳ ▲♦❣♦✱ ✉♠ sét✐♠♦ ❞❡ ✏❛❤❛✑ é1❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ✏❛❤❛✑ ❡ ✉♠ sét✐♠♦

❞❡ ✏❛❤❛✑ é 7 + 1 = 8 ₆= 19✳ ❊♥tã♦✱ ♦ ❢❛t♦r ❞❡ ❝♦rr❡çã♦ é ✉♠ ♥ú♠❡r♦ q✉❡ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r 8 é ✐❣✉❛❧ ❛ 19✱ ♦✉ s❡❥❛✱ 19

8 ✳ P♦rt❛♥t♦✱ ♦ ✈❛❧♦r ❞❡ ✏❛❤❛✑ é 19

8 ·7 = 133

8

■

❊ss❡ ♣r♦❜❧❡♠❛ tr❛♥s❝r✐t♦ ♣❛r❛ ❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♠♦❞❡r♥❛ s❡r✐❛✿

Problema 24:

❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r q✉❡ s♦♠❛❞♦ ❛ s✉❛ sét✐♠❛ ♣❛rt❡ é ✐❣✉❛❧ ❛ ✶✾✳

Solução:

✶ _{✭■♠❛❣❡♠ ❝♦♣✐❛❞❛ ❞♦ ❣♦♦❣❧❡✮ ❉✐❝❛✿ ❛ss✐st❛ ❛♦ ✜❧♠❡ ❆❧❡①❛♥❞r✐❛}

(43)

✷✳✷✳ ❖ ❈á❧❝✉❧♦ ❆❧❣é❜r✐❝♦ ✹✶

❙❡❥❛ x ♦ ✈❛❧♦r ♣r♦❝✉r❛❞♦✱ ♦✉ s❡❥❛✱ ♦ ✏❛❤❛✑ ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✳ ❊s❝r❡✈❡♥❞♦ ♦ ❡♥✉♥❝✐❛❞♦ ♥❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ❛t✉❛❧✱ t❡♠♦s✿ x+ 1

7x= 19✳ ▲♦❣♦✱

x+1

7x= 19 ⇔

x+1

7x= 19

·7

⇔ 7x+ 1x= 133

⇔ 8x= 133

⇔ (8x= 133)_÷8

⇔ x= 133 8

■

❖❜s❡r✈❡♠♦s q✉❡ ♦ ♣r♦❜❧❡♠❛ s❡ r❡s✉♠❡ ❡♠ ❡♥❝♦♥tr❛r ❛ r❛✐③ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❛❧❣é✲ ❜r✐❝❛

x+1

7x= 19

✱ ♦✉ s❡❥❛✱ ♦ ✈❛❧♦r x♣❛r❛ q✉❡ ♦ ♣♦❧✐♥ô♠✐♦P(x) = x+1

7xt❡♥❤❛ ❝♦♠♦

✈❛❧♦r ♥✉♠ér✐❝♦ ✶✾(P(x) = 19)✳

❖ ♣❛♣✐r♦ ❞❡ ❆❤♠❡s ♦✉ ❘❤✐♥❞✱ ❝♦♠♦ é ♠❛✐s ❝♦♥❤❡❝✐❞♦✱ é ♦ ❞♦❝✉♠❡♥t♦ q✉❡ ♠❛r❝❛ ❛ ♦r✐❣❡♠ ❞❛ á❧❣❡❜r❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♦ s✉r❣✐♠❡♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ♥❛ ♥♦ss❛ ❤✐stór✐❛✱ ❥á q✉❡ ♦s ♣♦❧✐♥ô♠✐♦s ❢❛③❡♠ ♣❛rt❡ ❞❛ á❧❣❡❜r❛✳ ❆❣♦r❛✱ q✉❡ t❛❧ ❛♣r❡♥❞❡r♠♦s ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ á❧❣❡❜r❛✱ ♦✉ ♠❡❧❤♦r✱ ❞❛ á❧❣❡❜r❛ ❞♦s ♣♦❧✐♥ô♠✐♦s❄ ▼❛s ♦ q✉❡ é ✉♠ ♣♦❧✐♥ô♠✐♦❄ ❊ss❛ r❡s♣♦st❛ ✈❛♠♦s t❡r ❡♠ ❜r❡✈❡✳

✷✳✷ ❖ ❈á❧❝✉❧♦ ❆❧❣é❜r✐❝♦

▼❛✐s ✉♠❛ ❤✐stór✐❛❄ ❙✐♠✦ ◗✉❡ ót✐♠♦✳ ❈♦♥t✐♥✉❡♠♦s✳ ✸

❋✐❣✉r❛ ✷ ✕ ❙✐♠♦♥ ❙t❡✈✐s

❙✐♠♦♥ ❙t❡✈✐s ✭✶✺✹✽✲✶✻✷✵✮ ♥❛s❝❡✉ ❡♠ ❇r✉❣✉❡s✱ ❢♦✐ ❝♦♠❡r❝✐❛♥t❡ ❡♠ ❆♥t✉ér♣✐❛✱ ✈✐❛❥♦✉ ♣❡❧❛ ❉✐♥❛♠❛r❝❛ ❡ ♣❛r❛ ❢♦r❛ ❞❛ ❊✉r♦♣❛✱ ❡ ❛♦s ✸✺ ❛♥♦s ✐♥❣r❡ss♦✉ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ▲❡②❞❡♥✱ ♦♥❞❡ ❡st✉❞♦✉ ♠❛t❡♠át✐❝❛✱ ❣r❡❣♦ ❡ ❡♥❣❡♥❤❛✲ r✐❛✳

❆♣❡s❛r ❞❡ t❡r ❡♥tr❛❞♦ ❝♦♠ ✐❞❛❞❡ ❢♦r❛ ❞♦ ♣❛❞rã♦ ✭♣♦r ♠♦t✐✈♦s ❢❛♠✐❧✐❛r❡s✮✱ ❝♦♥t✐♥✉♦✉ ❝♦♠♦ ♣r♦❢❡ss♦r ❞✉✲ r❛♥t❡ ❛❧❣✉♥s ❛♥♦s✱ t♦r♥❛♥❞♦✲s❡ ❛♠✐❣♦ ❞❡ ✉♠ ❛♣❧✐❝❛❞♦