❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás
❘❡❣✐♦♥❛❧ ❈❛t❛❧ã♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❖ ▼ét♦❞♦ ❞❡ ❈❛r❞❛♥♦ ❡ ❙✉❛ ❆♣❧✐❝❛çã♦ ♥♦
❊♥s✐♥♦ ▼é❞✐♦
❈❧❛✉❞✐♦ ❯♠❜❡rt♦ ❞❡ ▼❡❧♦
❈❛t❛❧ã♦
❈❧❛✉❞✐♦ ❯♠❜❡rt♦ ❞❡ ▼❡❧♦
❖ ▼ét♦❞♦ ❞❡ ❈❛r❞❛♥♦ ❡ ❙✉❛ ❆♣❧✐❝❛çã♦ ♥♦
❊♥s✐♥♦ ▼é❞✐♦
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❘❡❣✐♦✲ ♥❛❧ ❈❛t❛❧ã♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ P♦r❢ír✐♦ ❆③❡✈❡❞♦ ❞♦s ❙❛♥t♦s ❏ú♥✐♦r✳
❈❛t❛❧ã♦
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s✱ ♣❡❧❛ ✈✐❞❛ ❡ ♣♦r t♦❞♦s ♦s ❞♦♥s ❞❛❞♦s ❛ ♠✐♠✳
❆♦s ♠❡✉s ♣❛✐s✱ ♣❡❧♦s ❡①❡♠♣❧♦s ❡ ♣❡❧♦ ❝❛r✐♥❤♦ ❞❡❞✐❝❛❞♦ ❛ ♠✐♠✳
❆♦s ♠❡✉s ✜❧❤♦s✱ ❘❡❣✐♥❛❧❞♦✱ ❆♥❛ ❈❧❛r❛ ❡ ●✉✐❧❤❡r♠❡✱ ♣❡❧♦s q✉❛✐s ❡♥❝♦♥tr♦ ❢♦rç❛s ♣❛r❛ ✈❡♥❝❡r ♠❛✐s ✉♠ ❞✐❛✳
❆ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ s♦❜r❡ ♦ ♠ét♦❞♦ ❞❡ ❈❛r❞❛♥♦ ❛♣❧✐❝❛❞♦ ❡♠ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞♦ ✸♦ ❣r❛✉ ❞❛ ❢♦r♠❛ x3
+px+q= 0, p, q ∈R ❡ ❛ ✉t✐❧✐③❛çã♦ ❡♠ s❛❧❛✱ ❞♦ ✸♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ♦ ♣r♦❝❡❞✐♠❡♥t♦ s❡♠ ♦ ✉s♦ ❞❡ ❢ór♠✉❧❛ ♣❛r❛ ❞❡✲ t❡r♠✐♥❛r ✉♠❛ r❛✐③ ❞❡ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞♦ ✸♦ ❣r❛✉✳ ❆ ❛♣❧✐❝❛çã♦ ❞❡st❡ ♠ét♦❞♦ ❜✉s❝❛
♣♦ss✐❜✐❧✐t❛r ❛♦s ❛❧✉♥♦s ✉♠ ❡♥r✐q✉❡❝✐♠❡♥t♦ ✐♥t❡❧❡❝t✉❛❧ r❡❧❡✈❛♥t❡ ♣❛r❛ ❢✉t✉r♦s ❡st✉❞♦s ❞❛s ❝✐ê♥❝✐❛s ❡①❛t❛s✳ ◆❡st❡ tr❛❜❛❧❤♦ ♥ã♦ ❢♦✐ ✉t✐❧✐③❛❞♦ ✉♠❛ ❛✈❛❧✐❛çã♦ ❞✐❛❣♥óst✐❝❛ ♣❛r❛ ❛♥❛❧✐s❛r ♦ ♥í✈❡❧ ❞❡ ❝♦♠♣r❡❡♥sã♦ ❞♦ t❡♠❛✱ ❛♣❡♥❛s ❜✉s❝♦✉ ❛♣❧✐❝❛r ♦ ♣r♦❝❡❞✐♠❡♥t♦ ✉t✐✲ ❧✐③❛❞♦ ♣♦r ❈❛r❞❛♥♦✱ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ✉♠❛ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛ ❣❡r❛❧ ❞♦ ✸♦ ❣r❛✉✳ ❖ ❡st✉❞♦ tr❛③ ✉♠❛
❛❜♦r❞❛❣❡♠ ❤✐stór✐❝❛ ❞❛s r❡s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ✉♠❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ♣♦❧✐♥ô♠✐♦s✱ ❞❡st❛❝❛♥❞♦ ♦s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s✱ ♣r♦♣♦s✐çõ❡s ❡ ❞❡✜♥✐çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳ ❆❧é♠ ❞✐ss♦✱ ❞❡st❛❝❛ ♦ ❡st✉❞♦ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ ✸♦ ❣r❛✉ ❞❡ ❢♦r♠❛ ❛♥❛❧ít✐❝❛ ❡
❣rá✜❝❛✱ ♦♥❞❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ r❡s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞♦ ✹♦ ❣r❛✉✳ ❈♦♥✲
t✉❞♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡st❡ ❡st✉❞♦ ❞❡♠♦♥str❛ q✉❡ ♦s ❛❧✉♥♦s ❛♣r❡s❡♥t❛♠ ♠❛✐♦r ❢❛❝✐❧✐❞❛❞❡ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ r❛✐③ ❞❡ ✉♠❛ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛ ❣❡r❛❧✱ ❛ss✐♠ ❝♦♠♦ ❛s ❞❡♠❛✐s r❛í③❡s✳ P♦rt❛♥t♦✱ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ✉t✐❧✐③❛❞♦ ❡♠ s❛❧❛ ❛♣r❡s❡♥t❛ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r ♣❡❧♦ ♠❡♥♦s ✉♠❛ r❛✐③ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ ✸♦ ❣r❛✉✱ s❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡
❢ór♠✉❧❛✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ♣r❡s❡♥ts ❛ st✉❞② ♦♥ t❤❡ ❈❛r❞❛♥♦✬s ♠❡t❤♦❞ ❛♣♣❧✐❡❞ ✐♥ ✸r❞ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠x3
+px+q = 0, p, q ∈R ❛♥❞ ✉s❡ ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ t❤❡ ✸r❞ ②❡❛r ♦❢ ❤✐❣❤ s❝❤♦♦❧✱ ✇♦r❦✐♥❣ ✇✐t❤ t❤❡ ♣r♦❝❡❞✉r❡ ✇✐t❤♦✉t t❤❡ ✉s❡ ♦❢ ❢♦r♠✉❧❛ t♦ ❞❡t❡r♠✐♥❡ ❛ r♦♦t ♦❢ ✸r❞ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥✳ ❚❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤✐s ♠❡t❤♦❞ s❡❛r❝❤❡s ❡♥❛❜❧❡ t♦ st✉❞❡♥ts ❛ r❡❧❡✈❛♥t ✐♥t❡❧❧❡❝t✉❛❧ ❡♥r✐❝❤♠❡♥t ❢♦r ❢✉t✉r❡ st✉❞✐❡s ♦❢ t❤❡ ❡①❛❝t s❝✐❡♥❝❡s✳ ■♥ t❤✐s ✇♦r❦ ♥♦t ✉s❡❞ ❛ ❞✐❛❣♥♦st✐❝ ❡✈❛❧✉❛t✐♦♥ t♦ ❛♥❛❧②s❡ t❤❡ ❧❡✈❡❧ ♦❢ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡♠❡✱ ♦♥❧② s❡❛r❝❤ t♦ ❛♣♣❧② t❤❡ ♣r♦❝❡❞✉r❡ ✉s❡❞ ❜② ❈❛r❞❛♥♦✱ ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ ❛♥❞ ❡s♣❡❝✐❛❧❧② ♣r❡s❡♥t ❛ ❞❡♠♦♥str❛t✐♦♥ ♦❢ t❤✐s ♣r♦❝❡❞✉r❡ t♦ ❛♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ t❤❡ ✸r❞ ❞❡❣r❡❡✳ ❚❤❡ st✉❞② ❜r✐♥❣s ❛ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤ ♦❢ t❤❡ r❡s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥s✱ ❛❢t❡r✱ ❛ t❤❡♦r❡t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥ ❢♦r t❤❡ st✉❞② ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ ❞❡t❛❝❤✐♥❣ t❤❡ t❤❡♦r❡♠s ♠❛✐♥✱ ♣r♦♣♦s✐t✐♦♥s ❛♥❞ ❦❡② ❞❡✜♥✐t✐♦♥s ❢♦r t❤❡ st✉❞② ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s✳ ▼♦r❡♦✈❡r✱ ❞❡t❛❝❤ t❤❡ st✉❞② ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ r♦♦ts ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✸r❞ ❞❡❣r❡❡ ♦❢ ❛♥❛❧②t✐❝❛❧ ❛♥❞ ❣r❛♣❤✐❝❛❧ ❢♦r♠✱ ✇❤❡r❡ ✇❡ ♣r❡s❡♥t ❛♥ ❛♥❛❧②t✐❝❛❧ r❡s♦❧✉t✐♦♥ ❢♦r t❤❡ ✹t❤ ❞❡❣r❡❡ ❡q✉❛t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤✐s st✉❞② ❞❡♠♦♥str❛t❡s t❤❛t st✉❞❡♥ts ❤❛✈❡ ❣r❡❛t❡r ❢❛❝✐❧✐t② t♦ ✜♥❞ ❛ r♦♦t ♦❢ ❛♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✱ ❛s ✇❡❧❧ ❛s t❤❡ ♦t❤❡r r♦♦ts✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❝❡❞✉r❡ ✉s❡❞ ✐♥ ❝❧❛ssr♦♦♠ ♣r❡s❡♥ts ❛ ♠❡t❤♦❞ t♦ ✜♥❞ ❛t ❧❡❛st ♦♥❡ r♦♦t ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✸r❞ ❞❡❣r❡❡✱ ✇✐t❤♦✉t t❤❡ ✉s❡ ♦❢ ❢♦r♠✉❧❛✳
❑❡②✇♦r❞s
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ●rá✜❝♦ ❞❛s ❢✉♥çõ❡s f(x) =x3
+ 3x2
+ 2x+ 3 ❡ h(s) =s3
−s+ 3 ✳ ✻✷
✷ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) =x3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
✸ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) =x3+ 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
✹ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ h(s) =s3
−1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
✺ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) =x3
+x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✻ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ h(s) =s3
−s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✼ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) =x3+x+ 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✽ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ h(s) =s3
−s+ 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✾ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) =x3−3x+ 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✶✵ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ h(s) =s3−3s+ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
❙✉♠ár✐♦
✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s ✶✺
✷ Pr❡❧✐♠✐♥❛r❡s ✶✽
✷✳✶ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❆♥❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹ ❆❧❣♦r✐t♠♦ Prát✐❝♦ ❞❛ ❉✐✈✐sã♦ ❞❡ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ❘❛✐③❡s ❞❡ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✻ ❆❧❣♦r✐t♠♦ ❞❡ ❍♦r♥❡r ✲ ❘✉✜♥♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✼ ❉❡r✐✈❛❞❛ ❞❡ ✉♠ P♦❧✐♥ô♠✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✸✾
✸✳✶ ▼ét♦❞♦s ♣❛r❛ ❘❡s♦❧✈❡r ✉♠❛ ❊q✉❛çã♦ P♦❧✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✶✳✶ ❊q✉❛çã♦ ❞♦ 1♦ ●r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✶✳✷ ❊q✉❛çã♦ ❞♦ 2♦ ●r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✶✳✸ ❊q✉❛çã♦ ❞♦ 3♦ ●r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸✳✶✳✹ ❖ ▼ét♦❞♦ ❞❡ ❈❛r❞❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✶✳✺ ❊q✉❛çã♦ ❞❛ ❋♦r♠❛ x3
+px=q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✶✳✻ ❊q✉❛çã♦ ❞❛ ❋♦r♠❛ x3
=px+q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✶✳✼ ❊q✉❛çã♦ ❞❛ ❋♦r♠❛ x3
+px+q= 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✸✳✶✳✽ ❊q✉❛çã♦ ❞❛ ❋♦r♠❛ ●❡r❛❧ax3
+bx2
+cx+d= 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✸✳✶✳✾ ❈❛r❛❝t❡ríst✐❝❛s ❞❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ 3♦ ●r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✸✳✷ ❊st✉❞❛♥❞♦ ❛ ❈❛r❛❝t❡ríst✐❝❛ ❞❛s ❘❛í③❡s ❝♦♠ ❆✉①í❧✐♦ ❞❛ ❘❡♣r❡s❡♥t❛çã♦ ●rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✸ ❊q✉❛çã♦ ❞♦ 4♦ ●r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
■♥tr♦❞✉çã♦
❆ r❡❞✉çã♦ ❞❛ ❝❛r❣❛ ❤♦rár✐❛ ❞❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ♠❛t❡♠át✐❝❛ ♥❛ r❡❞❡ ♣ú❜❧✐❝❛ ❞❡ ❡♥s✐♥♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♦❝❛s✐♦♥❛❞❛ ♣❡❧❛ ✐♥s❡rçã♦ ❞❡ ♥♦✈❛s ❞✐s❝✐♣❧✐♥❛s ♥♦ ❝✉rrí✲ ❝✉❧♦ ❜ás✐❝♦✱ ❢❛③ ❝♦♠ q✉❡ ♦s ♣r♦❢❡ss♦r❡s s❡ ❛❞❡q✉❡♠ ❡♠ r❡❧❛çã♦ ❛ ❢♦r♠❛ ❞❡ ♠✐♥✐str❛r ♦s ❝♦♥t❡ú❞♦s ❞❡ ♠❛t❡♠át✐❝❛✳ ❉❡st❛ ❢♦r♠❛✱ ❡st❛ r❡❞✉çã♦ ❧❡✈❛ ♦s ♣r♦❢❡ss♦r❡s ❛ ♣r✐♦r✐③❛r❡♠ ❞❡t❡r♠✐♥❛❞♦s ❝♦♥t❡ú❞♦s ❛ s❡r❡♠ ♠✐♥✐str❛❞♦s✱ ❜❡♠ ❝♦♠♦✱ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❛ s❡r ✉t✐❧✐✲ ③❛❞❛✳ ❊st❛ s✐t✉❛çã♦ ❢❛③ ❝♦♠ q✉❡ s❡❥❛♠ ✉t✐❧✐③❛❞❛s✱ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❢ór♠✉❧❛s ♣r♦♥t❛s ❡ ♠❡❝❛♥✐③❛❞❛s✱ ❧❡✈❛♥❞♦ ♦ ❛❧✉♥♦ à ♣rát✐❝❛ ❞❛ ♠❡♠♦r✐③❛çã♦ ❞❡ ❢ór♠✉❧❛s✳
❆ ✈✐✈ê♥❝✐❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ♥❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s✱ ❞❡s✈❡❧♦✉ ❛ ❞✐✜❝✉❧❞❛❞❡ ❞♦s ❛❧✉♥♦s ❡♠ r❡s♦❧✈❡r ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳ ❊ss❛ ❞✐✜❝✉❧❞❛❞❡ ♠♦t✐✈♦✉ ❛ ❜✉s❝❛r ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ q✉❡ ❛✉①✐❧✐❡ ♦s ❛❧✉♥♦s ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ r❡s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳ ◆❛ ❜✉s❝❛ ❞❡ s❛♥❛r ❡ss❛s ❞✐✜❝✉❧❞❛❞❡s✱ ♣r♦♣♦♠♦s q✉❡ ♦ ❡st✉❞♦ ❞❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡✈❛ s❡r ❝♦♥str✉í❞♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❛❧✉♥♦✱ ❧❡✈❛♥❞♦✲♦ ❛ ❞❡s❝♦❜r✐r ❝♦♠♦ ❡ss❛s ❢ór♠✉❧❛s s✉r❣✐r❛♠ ❡ ♣♦rq✉❡ s✉r❣✐r❛♠✳
❆ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞♦♠✐♥❛r ✉♠❛ té❝♥✐❝❛ ♣❛r❛ s♦❧✉❝✐♦♥❛r ❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✈❡♠ ❞❡s❛✜❛♥❞♦ ♦s ♠❛t❡♠át✐❝♦s ❞❡s❞❡ ❛ ▼❡s♦♣♦tâ♠✐❛ q✉❛♥❞♦ ❢♦r❛♠ ❡♥❝♦♥tr❛❞♦s ♦s ♣r✐♠❡✐r♦s r❡❣✐str♦s ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❞♦ 1♦ ❡ 2♦ ❣r❛✉s✳ ❉❡s❞❡ ❡♥tã♦
❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❞✐✈❡rs❛s s✐t✉❛çõ❡s ♣r♦❜❧❡♠❛s✱ ♣r✐♥✲ ❝✐♣❛❧♠❡♥t❡ ❛s ❡q✉❛çõ❡s ❞❡ ❣r❛✉ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 3✱ ♣♦r ✐ss♦ ♠❡r❡❝❡ ✉♠ ❡st✉❞♦ ♠❛✐s
❛♣r♦❢✉♥❞❛❞♦✳
◗✉❛♥❞♦ ❞❡s❡♥✈♦❧✈❡♠♦s ♦ ❡st✉❞♦ ❞♦s ♣♦❧✐♥ô♠✐♦s✱ ❞❡♣❛r❛♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ q✉❡s✲ t✐♦♥❛♠❡♥t♦✿ q✉❛❧ é ❛ ♠❡❧❤♦r ❡str❛té❣✐❛ ♣❛r❛ tr❛❜❛❧❤❛r♠♦s ♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s❄ ❊st❛ é ✉♠❛ ✐♥❞❛❣❛çã♦ q✉❡ ♠✉✐t❛s ✈❡③❡s ✜❝❛ s❡♠ r❡s♣♦st❛✱ ♣♦✐s ❝♦♠ ♦ ♣❛ss❛r ❞♦s ❛♥♦s✱ ♦ ♥ú♠❡r♦ ❞❡ ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s ✈❡♠ s❡♥❞♦ r❡❞✉③✐❞❛s ❡ ❛❝❛❜❛ ♦❜r✐❣❛♥❞♦ ♦s ♣r♦❢❡ss♦r❡s ❛ ♣r✐♦r✐③❛r❡♠ ❞❡t❡r✲ ♠✐♥❛❞♦s ❝♦♥t❡ú❞♦s ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❛♣r❡s❡♥t❛r❡♠ ❢ór♠✉❧❛s ♣r♦♥t❛s✳ ❊ss❛ ♣rát✐❝❛ ❞❡ ❡♥s✐♥♦ ❝♦♥tr❛❞✐③ ♦s P❛râ♠❡tr♦s ◆❛❝✐♦♥❛✐s ❈✉rr✐❝✉❧❛r❡s ✭P❈◆✮ q✉❡ ♦r✐❡♥t❛ q✉❡ ♦ ❝♦♥❤❡✲ ❝✐♠❡♥t♦ ❞❡✈❛ s❡r ❝♦♥str✉í❞♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❛❧✉♥♦✱ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ♣❡rs♣❡❝t✐✈❛ só❝✐♦ ❤✐stór✐❝❛ ❞♦ t❡♠❛✳
❉✐❛♥t❡ ❞❡ss❛s ✐♥q✉✐❡t❛çõ❡s✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ♣r♦♣♦♠♦s ❡st✉❞❛r ✉♠❛ ♠❛♥❡✐r❛✱ q✉❡ ♣♦ss✐❜✐❧✐t❡ ❛♠❡♥✐③❛r ♦ ❡st✉❞♦ ❞❡ r❛í③❡s ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ✹✳ P❛r❛ t❛♥t♦ ❜❛s❡❛♠♦s ❡st❡ ❡st✉❞♦ ♥♦ ❛rt✐❣♦ ❞♦ Pr♦❢❡ss♦r ❊❧♦♥ ▲❛r❣❡s ▲✐♠❛ ❬✸❪✳ P♦rt❛♥t♦✱ ♦❜❥❡t✐✈❛♠♦s ❡st✉❞❛r ♦ ♣r♦❝❡❞✐♠❡♥t♦ t❡♦r✐③❛❞♦ ♣♦r ❈❛r❞❛♥♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ✉♠❛ r❛✐③ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡3♦ ❣r❛✉✳ ❆❧é♠ ❞✐ss♦ ❜✉s❝❛♠♦s ❝♦♥str✉✐r✱ ♣❛ss♦✲❛✲
♣❛ss♦✱ ❝♦♥❝❡✐t♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛s r❛í③❡s ❞❛s ❡q✉❛çõ❡s ❞❡ 1♦ ❛♦ 4♦ ❣r❛✉✳
◆❛s Pr❡❧✐♠✐♥❛r❡s✱ ✜③❡♠♦s ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❤✐stór✐❝♦ ❞♦s ♣r✐♠❡✐r♦s r❡❣✐str♦s ❞❛s r❡s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞♦ 1♦ ❣r❛✉ s✉r❣✐❞♦s ♥♦ ❊❣✐t♦ ❆♥t✐❣♦✳ ❆ ♣❛rt✐r ❞❛í ♦ ❞❡s❡♥✲
✈♦❧✈✐♠❡♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s✱ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❝♦♠♣❧❡①❛✱ ❢♦✐ ✐❞❡♥t✐✜❝❛❞♦ ♥♦s ❡s❝r✐t♦s ❝✉♥❡✐❢♦r♠❡s ❞❛ ❛♥t✐❣❛ ▼❡s♦♣♦tâ♠✐❛✱ q✉❡ só ❢♦r❛♠ ❞❡❝✐❢r❛❞♦s ♥♦ ❝♦♠❡ç♦ ❞♦ sé❝✉❧♦ ❳■❳ ♣♦r ●r♦t❡❢❡♥❞❡✳ ❙♦♠❡♥t❡ ♥♦ sé❝✉❧♦ ❳■■✱ ❝♦♠ ❛ tr❛❞✉çã♦ ❞♦ ❧✐✈r♦ ❞❡ ❆❧❦❤♦✇❛r✐③✐ ✭✼✽✵ ✲ ✽✺✵✮✱ q✉❡ ❛ á❧❣❡❜r❛ ❝♦♠❡ç♦✉ ❛ s❡ ❡①♣❛♥❞✐r ♣❡❧❛ ❊✉r♦♣❛✳ ❆♣ós ♦ sé❝✉❧♦ ❳❱■✱ ❝♦♠ ♦s ❡st✉❞♦s ❞❡ ❈❛r❞❛♥♦✱ ❛ á❧❣❡❜r❛ r❡t♦♠♦✉ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❝♦♠ ✐ss♦ ♦s ♠❛✲ t❡♠át✐❝♦s ♣❡r❝❡❜❡r❛♠ q✉❡ ♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ❡r❛♠ s✉✜❝✐❡♥t❡s✱ ♥❛s❝❡♥❞♦✱ ❡♥tã♦✱ ❛s ♣r✐♠❡✐r❛s ✐❞❡✐❛s ❞❛ ❝r✐❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ P♦r ✜♠✱ ❞❡✜♥✐♠♦s ❝♦♥❝❡✐t✉❛❧♠❡♥t❡ ❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s s♦❜r❡ ✉♠ ❛♥❡❧ ❞❡ ✐♥t❡❣r✐❞❛❞❡K✱ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❍♦r♥❡r✲❘✉✜♥♥✐ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛s r❛í③❡s ❞❛s ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳
◆❛ ❙❡çã♦✱ ❊q✉❛çõ❡s P♦❧✐♥♦♠✐❛✐s✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡s♦❧✉çã♦ ❞❛s ❡q✉❛✲ çõ❡s ❞❡ ✶♦ ❛♦ ✹♦ ❣r❛✉✱ s❡♥❞♦ ♦ ❡♥❢♦q✉❡ ❞❡t❡r♠✐♥❛r ❛s r❛í③❡s ❞❡s❞❡ ❛s ❡q✉❛çõ❡s ❞❡ ✶♦ ❡
✷♦ ❣r❛✉✱ s❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❢ór♠✉❧❛s✳ ◆❛s ❡q✉❛çõ❡s ❞❡ ✸♦ ❣r❛✉✱ ❛♣❧✐❝❛♠♦s ✉♠❛ s✉❜s✲
t✐t✉✐çã♦ ❞❡ ✈❛r✐á✈❡❧ ❡ r❡❞✉③✐♠♦s à ❢♦r♠❛x3
+px+q = 0, (∗) p, q ∈N ❡ ❛♣❧✐❝❛♠♦s ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❈❛r❞❛♥♦ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ r❛✐③✳ ❊♠ s❡❣✉✐❞❛✱ ❞❡t❡r♠✐♥❛♠♦s ❛s ❞❡♠❛✐s r❛í③❡s ❡ s❡ ❡ss❛s sã♦ r❡❛✐s ♦✉ ❝♦♠♣❧❡①❛s✳ ❆♣r❡s❡♥t❛♠♦s✱ t❛♠❜é♠✱ ❛tr❛✈és ❞❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛❞❛ à ❡q✉❛çã♦(∗)✱ ✉♠❛ ♠❛♥❡✐r❛ ♣❛r❛
q✉❡ ♦ ❛❧✉♥♦ ♣♦ss❛ ♣❡r❝❡❜❡r✱ ❞❡ ❢♦r♠❛ ❝♦♥❝r❡t❛✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ♦✉ ♠❛✐s r❛í③❡s r❡❛✐s✳
◆❛s ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦s r❡s✉❧t❛❞♦s ❞♦ ♣r♦❝❡❞✐♠❡♥t♦ ❛♣❧✐❝❛❞♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❞❡♠♦♥str❛♠ q✉❡ é ♣♦ssí✈❡❧ tr❛❜❛❧❤❛r ❝♦♠ ❡q✉❛çõ❡s ❞❡ ✸♦ ❣r❛✉ ❞❡ ✉♠❛
❢♦r♠❛ ❣❡r❛❧✱ ♥ã♦ só ♥♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❝♦♥❢♦r♠❡ ♦r✐❡♥t❛♠ ♦s P❈◆✬s✳ ❖s r❡s✉❧t❛❞♦s ♠♦str❛♠✱ t❛♠❜é♠✱ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠ ❛✉♠❡♥t♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ♣❛r❛ q✉❡ ♦s ❝♦♥t❡ú❞♦s ❞❡ ▼❛t❡♠át✐❝❛ ♣♦ss❛♠ s❡r tr❛❜❛❧❤❛❞♦s s♦❜ ✉♠❛ ♣❡rs♣❡❝t✐✈❛ só❝✐♦ ❤✐stór✐❝❛✳
✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s
❖s ♣r✐♠❡✐r♦s r❡❣✐str♦s ❞♦s ♣♦❧✐♥ô♠✐♦s ❡♥❝♦♥tr❛❞♦s ♥♦s ♣❛♣✐r♦s ❞♦ ❊❣✐t♦ ❛♥t✐❣♦ ♣❡✲ ❞✐❛♠ ❛ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✱ ✧❞❛ ❢♦r♠❛ x +ax = b ♦✉ x +ax + bx = c✱ ♦♥❞❡ a, b ❡ c ❡r❛♠ ❝♦♥❤❡❝✐❞♦s ❡ x é ❞❡s❝♦♥❤❡❝✐❞♦✳ ❆ ✐♥❝ó❣♥✐t❛ é ❝❤❛♠❛❞❛ ❞❡ ✬❛❤❛✬✧✭❇❖❨❊❘✱ ✷✵✵✷✱ ♣✳ ✶✶✮✳ ❊ss❛s ❢♦r♠❛s ❞❡ ❝á❧❝✉❧♦s ❡r❛♠ ❡①❡r❝í❝✐♦s ♣❛r❛ ❥♦✈❡♥s ❡st✉❞❛♥t❡s ❡ ❡♠❜♦r❛ ❢♦ss❡♠ ✉s❛❞♦s ❞❡ ❢♦r♠❛ ♣rát✐❝❛✱ ♦s ❡s❝r✐❜❛s ❛s ✉t✐❧✐③❛✈❛♠ ❝♦♠♦ ❢♦r♠❛ ❞❡ ❡♥✐❣♠❛s ♦✉ r❡❝r❡❛çõ❡s ♠❛t❡♠át✐❝❛s✳ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❢♦r♠❛ ♠❛✐s ❝♦♠♣❧❡①❛ ❢♦✐ ✐❞❡♥t✐✜❝❛❞♦ ♥♦s ❡s❝r✐t♦s ❝✉♥❡✐❢♦r♠❡s✶ ❞❛ ❛♥t✐❣❛ ▼❡s♦♣♦tâ✲
♠✐❛✷✳ ❖s ♣♦✈♦s q✉❡ ❛❧✐ ❤❛❜✐t❛✈❛♠ ❡r❛♠ ❞❡♥♦♠✐♥❛❞♦s ❞❡ ❜❛❜✐❧ô♥✐♦s✳ ▼✉✐t♦s ❞♦s r❡✲
❣✐str♦s ♠❛t❡♠át✐❝♦s ❞♦s ❜❛❜✐❧ô♥✐♦s✱ ❡♥❝♦♥tr❛❞♦s ❡♠ ❝❡♥t❡♥❛s ❞❡ t❛❜❧❡t❛s ❞❡ ❜❛rr♦✱ só ❢♦r❛♠ ❞❡❝✐❢r❛❞♦s ♥♦ ❝♦♠❡ç♦ ❞♦ sé❝✉❧♦ ❳■❳ ♣♦r ●r♦t❡❢❡♥❞❡✱ ♠❛s s♦♠❡♥t❡ ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳❳ q✉❡ ❝♦♠❡ç❛r❛♠ ❛ ❛♣❛r❡❝❡r ❡①♣♦s✐çõ❡s s✉❜st❛♥❝✐❛✐s ❞❛ ♠❛t❡♠át✐❝❛ ♠❡s♦✲ ♣♦tâ♠✐❝❛✳ ◆❡ss❡ r❡❣✐str♦✱ ❢♦✐ ♦❜s❡r✈❛❞♦ q✉❡ ♦s ❜❛❜✐❧ô♥✐♦s ♥ã♦ ♣♦ss✉í❛♠ ❞✐✜❝✉❧❞❛❞❡s ♥❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛ ❝♦♠♣❧❡t❛✱ ♣♦✐s ❡❧❡s t✐♥❤❛♠ ❞❡s❡♥✈♦❧✈✐❞♦ ♦♣❡r❛çõ❡s ❛❧❣é❜r✐❝❛s ✢❡①í✈❡✐s✳ ❙❡❣✉♥❞♦ ❇♦②❡r✱ ✷✵✵✷✱ ♦s ❜❛❜✐❧ô♥✐♦s ♣♦❞✐❛♠ ✧tr❛♥s♣♦rt❛r t❡r♠♦s ❡♠ ✉♠❛ ❡q✉❛çã♦ s♦♠❛♥❞♦ ✐❣✉❛✐s ❛ ✐❣✉❛✐s✱ ❡ ♠✉❧t✐♣❧✐❝❛r ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r q✉❛♥t✐✲ ❞❛❞❡s ✐❣✉❛✐s ♣❛r❛ r❡♠♦✈❡r ❢r❛çõ❡s ♦✉ ❡❧✐♠✐♥❛r ❢❛t♦r❡s✧✳ ❈♦♠♦ ♦ ❛❧❢❛❜❡t♦ ♥ã♦ t✐♥❤❛ s✐❞♦ ✐♥✈❡♥t❛❞♦✱ ♥ã♦ ✉t✐❧✐③❛✈❛♠ ❧❡tr❛s ♣❛r❛ ❞❡s✐❣♥❛r q✉❛♥t✐❞❛❞❡s ❞❡s❝♦♥❤❡❝✐❞❛s✱ ♣♦ré♠ ♣❛❧❛✲ ✈r❛s ❝♦♠♦ ✧❝♦♠♣r✐♠❡♥t♦✧✱ ✧❧❛r❣✉r❛✧✱ ✧ár❡❛✧ ❡ ✧✈♦❧✉♠❡✧ s❡r✈✐❛♠ ♣❛r❛ ♦s ❝á❧❝✉❧♦s✱ ♠❡s♠♦ q✉❡ ✉s❛❞♦s ♥✉♠ s❡♥t✐❞♦ ❛❜str❛t♦✳
❈♦♠ ❛ ❡①♣❛♥sã♦ ❡ ♦ ❞♦♠í♥✐♦ ❞♦ ■♠♣ér✐♦ ❘♦♠❛♥♦ ♥❛ ❊✉r♦♣❛ ♦ ✐♥t❡r❡ss❡ ♣❡❧❛ ▼❛t❡✲ ♠át✐❝❛ ❛❝❛❜♦✉ s❡♥❞♦ r❡❞✉③✐❞♦✱ ♣♦r ✐ss♦ ♥ã♦ t❡♠♦s ❣r❛♥❞❡s ♠❛r❝♦s ❤✐stór✐❝♦s r❡❣✐str❛❞♦s ♥❡ss❛ ár❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❙♦♠❡♥t❡ ❛ ♣❛rt✐r ❞♦ sé❝✉❧♦ ❳■■✱ ❝♦♠ ❛ tr❛❞✉çã♦ ❞♦ ❧✐✈r♦ ❍✐s❛❜ ❛❧✲❥❛❜r ✇❛❧✲♠✉❣❛❜❛❧❛✱ ❞❡ ❆❧❑❤♦✇❛r❛③✐ ✭✼✽✵ ✲ ✽✺✵✮ ❞♦ ár❛❜❡ ♣❛r❛ ♦ ❧❛t✐♠✱ q✉❡ s❡ ♣♦❞❡ ❝♦♥s✐❞❡r❛r ❝♦♠♦ ♦ ✐♥í❝✐♦ ❞❛ á❧❣❡❜r❛ ♥❛ ❊✉r♦♣❛✱ ♠❡r❡❝❡♥❞♦ ♦ tít✉❧♦ ❞❡ ✧♣❛✐ ❞❛ á❧❣❡❜r❛✧✳
❖s ♣r✐♥❝✐♣❛✐s ♠❛t❡♠át✐❝♦s ❞❡ss❛ é♣♦❝❛ ✈✐❡r❛♠ ❞❛ ■tá❧✐❛ ❞❛s ❝✐❞❛❞❡s ❞❡ ●❡♥♦✈❛✱
✶❆ ❡s❝r✐t❛ ❝✉♥❡✐❢♦r♠❡ é ❝♦♥s✐❞❡r❛❞❛ ♦ ♠❛✐s ❛♥t✐❣♦ s✐st❡♠❛ ❞❡ ❡s❝r✐t❛✳ ❘❡❛❧✐③❛❞❛ ♣❡❧♦s ❜❛❜✐❧ô♥✐❝♦s✱
❡ss❛ ❢♦r♠❛ ❞❡ ❡s❝r✐t❛ ❡r❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r sí♠❜♦❧♦s ✐❝♦♥♦❣rá✜❝♦s q✉❡ ❡r❛♠ t❛❧❤❛❞♦s ❝♦♠ ❡st✐❧❡t❡s✱ ♥♦ ❢♦r♠❛t♦ ❞❡ ❝✉♥❤❛✱ ❡♠ t❛❜❧❡t❛s ❞❡ ❜❛rr♦ ♠♦❧❡✱ q✉❡ ❡r❛♠ ❝♦❧♦❝❛❞❛s✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ❛♦ s♦❧ ♦✉ ❡♠ ❢♦r♥♦s ♣❛r❛ s❡❝❛r❡♠✳
✷❆ ▼❡s♦♣♦tâ♠✐❛ ❡r❛ ✉♠❛ r❡❣✐ã♦ ❧♦❝❛❧✐③❛❞❛ ❡♥tr❡ ♦s r✐♦s ❚✐❣r❡ ❡ ❊✉❢r❛t❡s✱ q✉❡ ❝♦♠♣r❡❡♥❞❡ ♦ ❛t✉❛❧
■r❛q✉❡
P✐s❛✱ ▼✐❧ã♦✱ ❋❧♦r❡♥ç❛ ❡ ❱❡♥❡③❛✱ ♣r✐♥❝✐♣❛✐s ❝❡♥tr♦s ❝♦♠❡r❝✐❛✐s ❡ ♦♥❞❡ ❝♦♠❡ç❛r❛♠ ❛ s✉r❣✐r tr❛❜❛❧❤♦s ✐♠♣♦rt❛♥t❡s ♥♦ ❞♦♠í♥✐♦ ❞❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s✳ ◆❡ss❛ é♣♦❝❛ ✈✐✈❡✉ ✉♠ ❞♦s ♠❛✐♦r❡s ♠❛t❡♠át✐❝♦s ❞❛ ❊✉r♦♣❛ ❞❛ ■❞❛❞❡ ▼é❞✐❛✿ ▲❡♦♥❛r❞♦ ❞❡ P✐s❛ ✭✶✶✼✺✲ ✶✷✺✵✮ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❋✐❜♦♥❛❝❝✐✳ ❊♠ ✶✷✵✷ ❡s❝r❡✈❡✉ s❡✉ ♠❛✐s ❢❛♠♦s♦ ❧✐✈r♦✱ ▲✐❜❡r ❆❜❛❝✐✳ ❊ss❛ ♦❜r❛ ❛♣r❡s❡♥t♦✉ q✉❡stõ❡s r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❡q✉❛çõ❡s ❞♦ ✷♦ ❡ ✸♦ ❣r❛✉ q✉❡
♦ ❛✉t♦r ❛♣r❡♥❞❡✉ ❝♦♠ ❛✉t♦r❡s ár❛❜❡s✱ ❛ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❞❛s ❡q✉❛çõ❡s ❡st✉❞❛❞❛s ❢♦✐ x3
+ 2x2
+ 10x = 20✳ ❆té ♦ sé❝✉❧♦ ❳❱■ ♥ã♦ ❤♦✉✈❡ ❣r❛♥❞❡s ❛❝♦♥t❡❝✐♠❡♥t♦s ♣❛r❛ ❛
❡✈♦❧✉çã♦ ❞❛ á❧❣❡❜r❛✳ ❙♦♠❡♥t❡ ❡♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ♠❡♥❝✐♦♥❛❞♦ q✉❡ ❤♦✉✈❡ ✉♠ ❛✈❛♥ç♦ ♥❛ ❞❡s❝♦❜❡rt❛ ❞❡ ❢ór♠✉❧❛s ❛❧❣é❜r✐❝❛s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ 3♦ ❡ 4♦ ❣r❛✉✱
♣♦ré♠ ❝♦♠ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❣❡♦♠étr✐❝❛ ❡ ✈❡r❜❛❧✳
❙❡❣✉♥❞♦ ❉♦♠✐♥❣✉❡s ❡ ■❡③③✐✱ ✷✵✵✸✱ ❡♠ ✶✺✾✶ ♦ ❢r❛♥❝ês ❋r❛♥ç♦✐s ❱✐èt❡ ✭✶✺✹✵ ✲ ✶✻✵✸✮✱ ❡♠ s✉❛ ♦❜r❛ ■♥tr♦❞✉çã♦ à ❆rt❡ ❆♥❛❧ít✐❝❛✱ ❝r✐♦✉ ♦ ❝á❧❝✉❧♦ ❧✐t❡r❛❧✱ ✐♥tr♦❞✉③✐♥❞♦ ❛ ❧✐♥❣✉❛✲ ❣❡♠ ❞❛s ❢ór♠✉❧❛s ♥❛ ♠❛t❡♠át✐❝❛✳ P❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❢♦✐ ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ❣❡♥❡r✐❝❛♠❡♥t❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✳ ❆♣❡s❛r ❞♦ ✉s♦ ❞❡ ✈♦❣❛✐s ❡ ❝♦♥s♦❛♥t❡s ♣❛r❛ r❡♣r❡s❡♥t❛r ✈❛r✐á✈❡✐s ❡ ❛s ❝♦♥st❛♥t❡s ♥ã♦ t❡r ✈✐♥❣❛❞♦✱ ❢♦✐ ✉♠❛ r❡✈♦❧✉çã♦ ♥❛ ♠❛t❡♠át✐❝❛✳ ❍♦❥❡ ♦ ✉s♦ ❞❡ ❝♦♥st❛♥t❡s ❧❡tr❛s ♥♦s ❝á❧❝✉❧♦s ♠❛t❡♠át✐❝♦s é ❞❡ ♣rá①✐s ❝♦♠✉♠✳ ❋♦✐ t❛♠❜é♠ ♥♦ sé❝✉❧♦ ❳❱■✱ ♥❛ ■tá❧✐❛✱ q✉❡ ❛ ❡✈♦❧✉çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ á❧❣❡❜r❛ ❣❛♥❤♦✉ ✉♠ ✐♠♣♦rt❛♥t❡ ❝♦♠♣♦♥❡♥t❡✳
P♦r ✈♦❧t❛ ❞❡ ✶✺✶✵✱ ❙❝✐♣✐♦♥❡ ❉❡❧ ❋❡rr♦ ✭✶✹✻✺ ✲ ✶✺✷✻✮✱ ♣r♦❢❡ss♦r ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇♦❧♦♥❤❛✱ ♣❡rs♦♥❛❣❡♠ s♦❜r❡ ❝✉❥❛ ✈✐❞❛ q✉❛s❡ ♥ã♦ s❡ s❛❜❡✱ ❡♥❝♦♥tr♦✉ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ r❡s♦❧✈❡r ❛s ❡q✉❛çõ❡s ❞♦ ✸♦ ❣r❛✉ ❞♦ t✐♣♦ x3
+cx+d= 0✳ ❉❡❧ ❋❡rr♦ ♥ã♦ ♣✉❜❧✐❝♦✉ ❛ s✉❛
❞❡s❝♦❜❡rt❛✱ ♣♦ré♠ r❡♣❛ss♦✉ ❛ ❞♦✐s ❞❡ s❡✉s ❞✐s❝í♣✉❧♦s✱ ❆♥♥✐❜❛❧❡ ❉❡❧❧❛ ◆❛✈❡ ✭♠❛✐s t❛r❞❡ s❡✉ ❣❡♥r♦ ❡ s✉❝❡ss♦r ♥❛ ❝❛❞❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡♠ ❇♦❧♦♥❤❛✮ ❡ ❆♥tô♥✐♦ ▼❛r✐❛ ❋✐♦r❡ ♦ s❡❣r❡❞♦ ❞❛ s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ ✧❝✉❜♦ ❡ ❝♦✐s❛s ✐❣✉❛❧ ❛ ♥ú♠❡r♦✧✭x3
+px=q✮ ❡ ✧❝✉❜♦ ✐❣✉❛❧ ❛ ❝♦✐s❛s ❡ ♥ú♠❡r♦✧✭x3
= px+q✮✳ P❛r❛ ❆♥tô♥✐♦ ▼❛r✐❛ ❋✐♦r❡ ❞❡✉ ❛ r❡❣r❛ ❡ ♥ã♦ ❛ ♣r♦✈❛✳ ❋✐♦r❡ ❛♣r♦♣r✐♦✉✲s❡ ❞❡st❡ ♠ér✐t♦✱ ❡ ❝♦♠ ❡st❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ ♠ã♦s r❡s♦❧✈❡✉ ❞❡s❛✜❛r ◆✐❝❝♦❧♦ ❚❛rt❛❣❧✐❛✸ ✭✶✹✾✾ ✲ ✶✺✺✼✮ ♣❛r❛ ✉♠ ✧❞✉❡❧♦ ✐♥t❡❧❡❝t✉❛❧✧
q✉❡ ❡r❛ ♠✉✐t♦ ❝♦♠✉♠ ♥❡st❛ é♣♦❝❛✱ ♦s q✉❛✐s ❡r❛♠ ♣r❡s✐❞✐❞♦s ♣♦r ❛❧❣✉♠❛ ❛✉t♦r✐❞❛❞❡ ❡ ♠✉✐t❛s ✈❡③❡s ❝♦♠ ✉♠❛ ❣r❛♥❞❡ ❛✉❞✐ê♥❝✐❛✳ ❊ss❛s ❞✐s♣✉t❛s ♠✉✐t❛s ✈❡③❡s ❞❡✜♥✐❛♠ ❛ ♣❡r✲ ♠❛♥ê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❛❧❣✉♥s ♣r♦❢❡ss♦r❡s ✉♥✐✈❡rs✐tár✐♦s ♥❛ ❝át❡❞r❛✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ◆✐❝❝♦❧♦ ❚❛rt❛❣❧✐❛ ❛❝❛❜♦✉ ❞❡s❝♦❜r✐♥❞♦ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ t✐♣♦ x3
+cx+d= 0 ❡ t❛♠❜é♠ ♣❛r❛ ♦ t✐♣♦ x3
+bx2
+d= 0 ♦ q✉❛❧ ❋✐♦r❡ ♥ã♦ ❝♦♥❤❡❝✐❛✳ ❊♠
s✉❛ ❞✐s♣✉t❛ ❋✐♦r❡ ♣r♦♣ôs ✸✵ ♣r♦❜❧❡♠❛s✱ t♦❞♦s ❡♥✈♦❧✈❡♥❞♦ ❛s ❡q✉❛çõ❡s ❞♦ t❡r❝❡✐r♦ ❣r❛✉
✸❈♦♥❝❡✐t✉❛❞♦ ♠❛t❡♠át✐❝♦ ❡ ♣r♦❢❡ss♦r ❡♠ ❱❡♥❡③❛ ❡ q✉❡ ❤❛✈✐❛ ❞❡rr♦t❛❞♦ ✈ár✐♦s ❞❡s❛✜❛♥t❡s✳
❡ ❚❛rt❛❣❧✐❛✱ ♣♦r s✉❛ ✈❡③✱ ♣r♦♣ôs ✉♠❛ ❧✐st❛ ❞❡ ♣r♦❜❧❡♠❛s ❞✐✈❡rs✐✜❝❛❞♦s✳ ❙❡❣✉♥❞♦ ❇♦②❡r✱ ✷✵✵✷✱ ❚❛rt❛❣❧✐❛ ♥ã♦ t✐♥❤❛ ♦ ❝♦st✉♠❡ ❞❡ ❞✐✈✉❧❣❛r ♦ ♠ét♦❞♦ ❞❡ r❡s♦❧✉çã♦ ❡ s✐♠ ❛♣❡♥❛s ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s✱ ♠❛s ❞❡st❛ ✈❡③✱ ❛❝❛❜♦✉ ❛♣r❡s❡♥t❛♥❞♦ ❡st❡ ♠ét♦❞♦ ❛ ✉♠ ❛♠✐❣♦ ●❡rô♥✐♠♦ ❈❛r❞❛♥♦ ✭✶✺✵✶ ✲ ✶✺✼✻✮✱ ✉♠ ❣r❛♥❞❡ ♠❛t❡♠át✐❝♦ q✉❡ ✈✐✈✐❛ ❡♠ ▼✐❧ã♦✱ ❛♦ q✉❛❧ ♣❡❞✐✉ q✉❡ ♥ã♦ ❞✐✈✉❧❣❛ss❡ ❡st❡ ♠ét♦❞♦✳ ❈♦♠ ❛ ❛❥✉❞❛ ❞♦ ❞✐s❝í♣✉❧♦ ▲✉❞♦✈✐❝♦ ❋❡rr❛r✐ ✭✶✺✷✷ ✲ ✶✺✻✺✮✱ ❈❛r❞❛♥♦ ❝♦♥s❡❣✉✐✉ ❞❡♠♦♥str❛r ♦ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❚❛rt❛❣❧✐❛ ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ t✐♣♦ x3
+cx=d✳ ❊♠ ✶✺✹✷✱ ❛♣ós ❛♥❛❧✐s❛r ♦s ♠❛♥✉s❝r✐t♦s ❞❡
❉❡❧ ❋❡rr♦✱ ❈❛r❞❛♥♦ ❡♥❝♦♥tr♦✉ ❛ ❢ór♠✉❧❛x= 3
s
−q2 +
r q2
4 +
p3
27+
3 s
−2q −
r q2
4 +
p3
27✱
❛ q✉❛❧ ❡r❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞❛ ❢♦r♠❛ x3
+px+q = 0, p, q ∈ N ❡ ❛ ♣✉❜❧✐❝♦✉✱ ❡♠ ✶✺✹✺✱ ❡♠ s❡✉ ❧✐✈r♦ ❆rs ▼❛❣♥❛✱ ✜❝❛♥❞♦ ❝♦♠ t♦❞♦ ♦ ❝ré❞✐t♦✳ ❆ ♣❛rt✐r ❞❛í ❢♦✐ ♣♦ssí✈❡❧ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ ✸♦ ❣r❛✉ ♥♦ ❝❛s♦ ❣❡r❛❧ax3
+bx2
+cx+d= 0❡ ❡q✉❛çõ❡s
❞♦ ✹♦ ❣r❛✉ ❞❛ ❢♦r♠❛ax4
+bx3
+cx2
+dx+e= 0✳
❆s r❡s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s s❡♠♣r❡ ❢❛s❝✐♥❛r❛♠ ♦s ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛✱ ♦s ❇❛❜✐❧ô♥✐♦s ❥á ❝♦♥s❡❣✉✐❛♠ r❡s♦❧✈❡r ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❞♦ ✷♦❣r❛✉ ❝♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛✲
❞♦s✱ ♦s ❣r❡❣♦s r❡s♦❧✈✐❛♠ ❛❧❣✉♠❛s ❡q✉❛çõ❡s t❛♠❜é♠ ❞♦ ✷♦ ❣r❛✉ ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✳
◆❛ ✐❞❛❞❡ ♠é❞✐❛ ❞✉r❛♥t❡ ❛ ❛s❝❡♥sã♦ ❞♦ ❈r✐st✐❛♥✐s♠♦✱ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛ s❡ ❞❡✈❡ ❛♦s ➪r❛❜❡s ❡ ♦s ❍✐♥❞ús✳ ❖s ❍í♥❞ús r❡❛❧✐③❛r❛♠ ♣❡sq✉✐s❛s ♥❛ ➪❧❣❡❜r❛ ❡ ♥♦ sé❝✉❧♦ ❳■ ♦ ♠❛t❡♠át✐❝♦ ❙r✐❞❤❛r❛ ❞❡s❡♥✈♦❧✈❡ ❛ ❢ór♠✉❧❛ x= −b±
√
b2
−4ac
2a q✉❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢ór♠✉❧❛ ❞❡ ❇❛s❑❛r❛✱ q✉❡✱ ❡♠ ❛❧❣✉♥s ❝❛s♦s✱ ♣♦❞❡r✐❛ s❡r r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ♥ú♠❡r♦ b2
−4ac < 0✱ ♠❛s ✐ss♦ ♥ã♦ ❡r❛ ✉♠ ♣r♦❜❧❡♠❛✱ ✈✐st♦ q✉❡✱ ♦ q✉❡ ✐♠♣♦rt❛✈❛✱ ❡r❛♠ ❡q✉❛çõ❡s
q✉❡ t✐♥❤❛♠ b2
−4ac >0✳
◆♦ sé❝✉❧♦ ❳❱■✱ r❡ss✉r❣✐✉ ♥❛ ❊✉r♦♣❛ ♦ ✐♥t❡r❡ss❡ ♣❡❧♦ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❡♠ ✉♠❛ ❞✐s♣✉t❛ ❡♥tr❡ ❈❛r❞❛♥♦ ❡ ❚❛rt❛❣❧✐❛ ♣❡❧❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ✸♦ ❣r❛✉✱ ♣❡r❝❡❜❡r❛♠
q✉❡ ♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ❡r❛♠ s✉✜❝✐❡♥t❡s✱ ♥❛s❝❡♥❞♦ ❡♥tã♦ ❛s ♣r✐♠❡✐r❛s ✐❞❡✐❛s ❞❛ ❝r✐❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
◆♦ sé❝✉❧♦ ❳❱■■✱ ❘❡♥é ❉❡s❝❛rt❡s ❡♠ ✉♠❛ ♣❛ss❛❣❡♠ ❞❡ s❡✉ ❧✐✈r♦ ❉✐s❝✉rs♦ ❙♦❜r❡ ♦ ▼ét♦❞♦ ❞❡ ❇❡♠ ❯t✐❧✐③❛r ❛ ❘❛③ã♦ ❡ ❞❡ ❊♥❝♦♥tr❛r ❛ ❱❡r❞❛❞❡ ❡s❝r❡✈❡✉ ❛ s❡❣✉✐♥t❡ ❢r❛s❡✿ ✧◆❡♠ s❡♠♣r❡ ❛s r❛í③❡s ✈❡r❞❛❞❡✐r❛s ✭♣♦s✐t✐✈❛s✮ ♦✉ ❢❛❧s❛s ✭♥❡❣❛t✐✈❛s✮ ❞❡ ✉♠❛ ❡q✉❛çã♦ sã♦ r❡❛✐s✳ ➪s ✈❡③❡s ❡❧❛s sã♦ ✐♠❛❣✐♥ár✐❛s✧✳
P♦r ❡ss❡ ♠♦t✐✈♦✱ ❛té ❤♦❥❡ ♦ ♥ú♠❡r♦ √−1 é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦✱ t❡r♠♦
q✉❡ s❡ ❝♦♥s❛❣r♦✉ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❡①♣r❡ssã♦ ✧♥ú♠❡r♦ ❝♦♠♣❧❡①♦✧✳ ▼❛s q✉❡♠ ❢❡③ ♦ tr❛❜❛❧❤♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❡ ❞❡❝✐s✐✈♦ s♦❜r❡ ♦ ❛ss✉♥t♦ ❢♦✐ ▲❡♦♥❛r❞♦ ❊✉❧❡r✱ ♦ q✉❛❧ ❢♦✐ ♥♦tá✈❡❧ ❡♠ s❡✉ ❡♠♣❡♥❤♦ ♥❛ ♠❡❧❤♦r✐❛ ❞❛ s✐♠❜♦❧♦❣✐❛✳ ❊s♣❡❝✐❛❧♠❡♥t❡✱ q✉❛♥❞♦✱ ❡st❛✈❛ ❡♠ ❡①♣❛♥sã♦ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ✐♥✈❡♥t❛❞♦ ♣♦r ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✳ ▼✉✐t❛s
❞❛s ♥♦t❛çõ❡s q✉❡ ✉t✐❧✐③❛♠♦s ❤♦❥❡ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ♣♦r ❡❧❡✳ ❉❡♥tr❡ ❛s r❡♣r❡s❡♥t❛çõ❡s ♣r♦♣♦st❛s ♣♦r ❊✉❧❡r ❞❡st❛❝❛♠♦s ♦is✉❜st✐t✉✐♥❞♦√−1✳ ❊✉❧❡r ♣❛ss♦✉ ❛ ❡st✉❞❛r ♥ú♠❡r♦s
❞❛ ❢♦r♠❛ z = a +bi✱ ♦♥❞❡ a, b ∈ R✱ ✉t✐❧✐③❛♥❞♦ i2
= −1 ♣❛r❛ ❡st❡ ❡st✉❞♦✳ ▼❛s✱
❡st❛ ✐❞❡✐❛ ✉t✐❧✐③❛❞❛ ♣♦r ❊✉❧❡r só ❢♦✐ ❛❝❡✐t❛ ❛ ♣❛rt✐r ❞♦s r❡s✉❧t❛❞♦ss ♣✉❜❧✐❝❛❞♦s ♣♦r ●❛✉ss ❛♣r❡s❡♥t❛♥❞♦ ✉♠❛ ♥♦✈❛ ❡str✉t✉r❛ ❞❡ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦✱ q✉❡ ❤♦❥❡ ❞❡♥♦♠✐♥❛♠♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❯♠ ❞♦s ❢❛t♦s q✉❡ ❧❡✈♦✉ ❛♦ s✉r❣✐♠❡♥t♦ ❞❡st❡ ♥♦✈♦ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦✱ ❢♦r❛♠ ❡q✉❛✲ çõ❡s ♣♦❧✐♥♦♠✐❛✐s q✉❡ ♥ã♦ ♣♦ss✉í❛♠ r❛í③❡s ♥♦ ✉♥✐✈❡rs♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ♣♦r ❡①❡♠♣❧♦✱ x2
+ 1 = 0✳
✷ Pr❡❧✐♠✐♥❛r❡s
❖s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s ♥❡st❛ s❡çã♦ s❡rã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦s ♣r✐♥❝✐♣❛✐s ❝❛s♦s ❡s✲ t✉❞❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳
✷✳✶
◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❉❡✜♥✐çã♦ ✶✳ ❯♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z é ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ (a, b)✱
♦♥❞❡ a, b ∈ R✱ ♦✉ ♥❛ ❢♦r♠❛ z = a +bi, ♦♥❞❡ i = √−1. ❊♠ t❡r♠♦s ❞❡ ♥♦t❛çã♦ ❞❡ ❝♦♥❥✉♥t♦ ✉s❛✲s❡ ❛ ❧❡tr❛C ♣❛r❛ ❞❡s✐❣♥❛r ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❛ ♥♦t❛çã♦ é✿
C={a+bi |a, b∈R ❡ i2
=−1}. ❈♦♥s✐❞❡r❡♠♦s ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✱z =a+bi✱ ♣♦❞❡♠♦s ❞❡st❛❝❛r✿
✭✶✮ ❆ ♣❛rt❡ r❡❛❧ ❞❡ z é r❡♣r❡s❡♥t❛❞❛ ♣♦r Re(z) = a ❡ ❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ z é r❡♣r❡s❡♥t❛❞❛ ♣♦rIm(z) =b❀
✭✷✮ ◗✉❛♥❞♦b = 0 ✐♠♣❧✐❝❛ ❡♠ z=a✱ ❞✐③❡♠♦s q✉❡ z é ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ r❡❛❧❀ ✭✸✮ ◗✉❛♥❞♦a= 0 ✐♠♣❧✐❝❛ ❡♠z =bi✱ ❞✐③❡♠♦s q✉❡ z é ✉♠ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦ ♣✉r♦❀ ✭✹✮ ◗✉❛♥❞♦a = 0 ❡ b= 1 ✐♠♣❧✐❝❛ ❡♠ z =i✱ ❞✐③❡♠♦s q✉❡ z é ❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❙❡❥❛♠ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s z = a+bi ❡ w = c+di✳ ❉❡✜♥✐♠♦s ❛ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♥t❡ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s z ❡ w✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r✿
z+w = (a+bi) + (c+di) = (a+c) + (b+d)i z.w = (a+bi)(c+di) = (ac−bd) + (ad+bc)i
❙❡❥❛ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z = a+bi✳ ❉❡✜♥✐♠♦s ♦ ♠ó❞✉❧♦ ❞♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ③✱ ❝♦♠♦ s❡♥❞♦
|z|=√a2+b2.
❉❡✜♥✐çã♦ ✷✳ ❙❡❥❛ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z r❡♣r❡s❡♥t❛❞♦ ♣♦r z =a+bi✱ ❝♦♠ a, b∈R✳ ❙❡✉ ❝♦♥❥✉❣❛❞♦ é ❞❛❞♦ ♣♦rz s❡♥❞♦ r❡♣r❡s❡♥t❛❞♦ ♣♦r z =a−bi✳
▲❡♠❛ ✶✳ ❙❡ z ❡ w sã♦ ❝♦♠♣❧❡①♦s ♥ã♦ ♥✉❧♦s q✉❛✐sq✉❡r✱ ❡♥tã♦✿ ✭❛✮ z ∈R⇔Im(z) = 0⇔z =z;
✭❜✮ z+w=z+w; zw =z.w; z
w
=
z w
;
✭❝✮ zn = (z)n; ✭❞✮|z|= 1⇔z = 1
z.
❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❙❡❥❛z =a+bi✳ ❚❡♠♦s
z ∈R⇔b= 0⇔a+bi =a−bi ⇔z =z.
✭❜✮ ❙❡♥❞♦z =a+bi✱ w=c+di✱ t❡♠♦s✿
z+w = (a−bi) + (c−di) = (a+c)−(b+d)i=z+w;
zw = (ac−bd) + (ad+bc)i= (ac−bd)−(ad+bc)i
= (a−bi)(c−di) =z.w;
z w
= z
w
.w w =
z w
.w w =
z w.
✭❝✮ ❙❡❣✉❡ ❞❡ ✭❜✮✱ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n✳
✭❞✮ ❙❡♥❞♦z.z =|z|2✱ t❡♠♦s q✉❡✿
|z|= 1⇔z.z = 1⇔z = 1
z.
✷✳✷ ❆♥❡❧
❉❡✜♥✐çã♦ ✸✳ ❯♠ s✐st❡♠❛ ♠❛t❡♠át✐❝♦ ❝♦♥st✐t✉í❞♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ K ❡ ✉♠ ♣❛r ❞❡ ♦♣❡r❛çõ❡s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✉♠❛ ❛❞✐çã♦ (x, y) 7→ x+y✱ ❡ ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦
(x, y)7→xy. ❈❤❛♠❛♠♦s (K,+, .) ❞❡ ❛♥❡❧ s❡✿
✐✮(K,+) é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✿
❛✮ s❡ a, b, c∈K✱ ❡♥tã♦ a+ (b+c) = (a+b) +c;
❜✮ s❡ a, b∈K✱ ❡♥tã♦ a+b=b+a;
❝✮ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ 0k∈K t❛❧ q✉❡✱ a+ 0k =a,∀a ∈K; ❞✮ ∀a∈K, ∃(−a)∈K✱ t❛❧ q✉❡ a+ (−a) = 0k;
✐✐✮ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛✿
s❡ a, b, c∈K✱ ❡♥tã♦ a(bc) = (ab)c.
✐✐✐✮ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ é ❞✐str✐❜✉t✐✈❛ ❡♠ r❡❧❛çã♦ à ❛❞✐çã♦✱ ✐st♦ é✿
s❡ a, b, c∈K✱ ❡♥tã♦ a(b+c) = ab+bc✳ ❙❡ ✉♠ ❛♥❡❧(K,+, .)s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡✿
✐✈✮ ∃1∈ K,06= 1, t❛❧ q✉❡ x.1 = 1.x= x. ∀x ∈K, ❡♥tã♦ ❞✐③❡♠♦s q✉❡ (K,+, .) é ✉♠
❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡ ✶✳
✈✮∀x, y ∈K, x.y =y.x✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ (K,+, .)é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✳
✈✐✮ ∀x, y ∈ K, x.y = 0 ⇒ x = 0 ♦✉ y = 0, ❡♥tã♦ ❞✐③❡♠♦s q✉❡ (K,+, .) é ✉♠ ❛♥❡❧
s❡♠ ❞✐✈✐s♦r❡s ❞❡ ③❡r♦✳
❙❡ K,+, . é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✱ ❝♦♠ ✉♥✐❞❛❞❡ ❡ s❡♠ ❞✐✈✐s♦r❡s ❞❡ ③❡r♦✱ ❞✐③❡♠♦s q✉❡
(K,+, .) é ✉♠ ❞♦♠í♥✐♦ ❞❡ ■♥t❡❣r✐❞❛❞❡✳
❙❡ ✉♠ ❞♦♠í♥✐♦ ❞❡ ■♥t❡❣r✐❞❛❞❡ (K,+, .) s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡✿
✈✐✐✮∀x∈K, x6= 0,∃y ∈K t❛❧ q✉❡ x.y =y.x= 1✱ ❞✐③❡♠♦s q✉❡ (K,+, .) é ✉♠ ❝♦r♣♦✳
❊①❡♠♣❧♦ ✶✳
✐✮ ❛♥❡❧ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s❀
✐✐✮ ❛♥❡❧ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s❀ ✐✐✐✮ ❛♥❡❧ ❞♦s ♥ú♠❡r♦s r❡❛✐s❀ ✐✈✮ ❛♥❡❧ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❊①❡♠♣❧♦ ✷✳ ❚♦❞♦s ♦s ❛♥é✐s ♥✉♠ér✐❝♦s✱Z✱ Q✱ R✱ C✱ sã♦ ❞♦♠í♥✐♦s ❞❡ ✐♥t❡❣r✐❞❛❞❡✳
Pr♦♣♦s✐çã♦ ✶✳ ❚♦❞♦ ❝♦r♣♦ é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛K ✉♠ ❝♦r♣♦ ❡ a, b∈K✱ t❛✐s q✉❡ a.b= 0✳
❙✉♣♦♥❤❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ a 6= 0✱ ❡ ♣♦rt❛♥t♦ ❛ é ✐♥✈❡rsí✈❡❧✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♦s ❞♦✐s
♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡ a.b= 0✱ ♣♦ra−1✿
a−1(ab) = a−1.0 = 0.
▼❛sa−1
(ab) =b✱ ❡♥tã♦ b= 0✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛b 6= 0 ♦ ❝❛s♦ é ❛♥á❧♦❣♦✳
❊♥tã♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❢❛t♦r❡s ❞❡K ♥ã♦ ♣♦❞❡ s❡r ♥✉❧♦ s❡♠ q✉❡ ✉♠ ❞❡❧❡s s❡❥❛ ♥✉❧♦✳ ▲♦❣♦ K é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳
✷✳✸
P♦❧✐♥ô♠✐♦s
◆❡st❡ tr❛❜❛❧❤♦ q✉❛♥❞♦ s❡ tr❛t❛r ❞❡ ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡ ✐♥✜♥✐t♦ ✉s❛♠♦sK ❡ q✉❛♥❞♦ r❡❧❛❝✐♦♥❛r ❛♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s s♦❜r❡ ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡ K ✉s❛♠♦sK[X]✳
❉❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛❧❣é❜r✐❝♦✱ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s r❡❛✐s ♣♦❞❡♠ s❡r ❡♥❝❛r❛❞❛s ❝♦♠♦ ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✳
❉❡✜♥✐çã♦ ✹✳ ❯♠❛ s❡q✉ê♥❝✐❛(a0, a1, a2,· · ·)❞❡ ❡❧❡♠❡♥t♦s ❞❡ K é ❞✐t❛ q✉❛s❡ t♦❞❛ ♥✉❧❛
s❡ ❡①✐st✐r n≥0 t❛❧ q✉❡
an=an+1 =· · ·= 0.
❊①❡♠♣❧♦ ✸✳ ❙ã♦ ❝♦♥s✐❞❡r❛❞❛s s❡q✉ê♥❝✐❛s q✉❛s❡ t♦❞❛ ♥✉❧❛✳
(0,0,0,0, ...) ❡ (1,2,3,4, ..., n,0,0,0, ...)✳
❉❡✜♥✐çã♦ ✺✳ ❯♠ ♣♦❧✐♥ô♠✐♦ s♦❜r❡ ✭♦✉ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠✮ K é ✉♠❛ ❡①♣r❡ssã♦ f =f(X)✱ ❝♦♠ f(X)∈K[X] ❞♦ t✐♣♦
f(X) = a0+a1X+a2X 2
+a3X 3
+· · ·=X
s≥0
asXs,
♦♥❞❡ (a0, a1, a2, ...) é ✉♠❛ s❡q✉ê♥❝✐❛ q✉❛s❡ t♦❞❛ ♥✉❧❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ K✳
❉❛❞♦ ✉♠ ♣♦❧✐♥ô♠✐♦ f(X) = a0 +a1X +a2X2 +a3X3 +· · · s♦❜r❡ K✱ ❛❞♦t❛♠♦s ❛s
s❡❣✉✐♥t❡s ❝♦♥✈❡♥çõ❡s✿
✐✮ ❖s ❡❧❡♠❡♥t♦s ai ∈K sã♦ ❞❡♥♦♠✐♥❛❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ f❀
✐✐✮ ◗✉❛♥❞♦ai = 0♦♠✐t✐♠♦s✱ s❡♠♣r❡ q✉❡ ❢♦r ❝♦♥✈❡♥✐❡♥t❡✱ ♦ t❡r♠♦aiXi✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♠♦ ❛ s❡q✉ê♥❝✐❛ (a0, a1, a3,· · ·)é q✉❛s❡ t♦❞❛ ♥✉❧❛✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ n≥0♣❛r❛
♦ q✉❛❧ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✱
f(X) =
n X
s=0
asXs;
✐✐✐✮ ◗✉❛♥❞♦ai =±1✱ ❡s❝r❡✈❡✲s❡±Xi✱ ❛♦ ✐♥✈és ❞❡(±1)Xi✱ ♣❛r❛ ♦ t❡r♠♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡ f❀
✐✈✮ ❖ ♣♦❧✐♥ô♠✐♦0 = 0 + 0X+ 0X2
+· · · é ❞❡♥♦♠✐♥❛❞♦ ♦ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡
♥✉❧♦ s♦❜r❡ K✳
▲❡♠❛ ✷✳ ❙❡ (as)s≥0 ❡ (bs)s≥0 sã♦ s❡q✉ê♥❝✐❛s q✉❛s❡ t♦❞❛s ♥✉❧❛s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ K✱
❡♥tã♦ t❛♠❜é♠ sã♦ q✉❛s❡ t♦❞❛s ♥✉❧❛s ❛s s❡q✉ê♥❝✐❛s (as±bs)s≥0 ❡ (cs)s≥0✱ ♦♥❞❡
cs = X
i+j=s
aibj, ❝♦♠ i, j ≥0.
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ m, n ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ t❛✐s q✉❡ ai = 0 ❡ bj = 0✱ ♣❛r❛ i > n❡ j > m✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥s✐❞❡r❡ s > m, n✱ ❛ss✐♠ as = 0 ❡bs = 0✱ ❧♦❣♦
(as±bs) = 0✳ P♦rt❛♥t♦✱ ❛ s❡q✉ê♥❝✐❛ (as±bs)s≥0 é ✉♠❛ s❡q✉ê♥❝✐❛ q✉❛s❡ t♦❞❛ ♥✉❧❛✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡s > m+n ❡ i+j =s✱ ❝♦♠ i, j ≥0✱ ♦♥❞❡ as ❡bs sã♦ s❡q✉ê♥❝✐❛s q✉❛s❡ t♦❞❛ ♥✉❧❛✱ ❡♥tã♦✿
P❛r❛ i > n❡ j > m✱ t❡♠♦s✿
ai = 0 ❡ bj = 0.
P❛r❛ i > n♦✉ j > m✱ t❡♠♦s✿
ai = 0 ♦✉ bj = 0.
❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ t❡♠♦s✿
cs = X
i+j=s
aibj = 0 ❝♦♠ i, j ≥0.
❉❡✜♥✐çã♦ ✻✳ ❉❛❞♦s ❡♠ K[X] ♦s ♣♦❧✐♥ô♠✐♦s
f(X) = X
s≥0
asXs e g(X) = X
s≥0
bsXs, as, bs∈K.
❆ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❡f ❡ g✱ ❞❡♥♦t❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r f+g ❡ f.g✳ ❆ss✐♠✱
(f+g)(X) =X
s≥0
asXs+ X
s≥0
bsXs = X
s≥0
(as+bs)Xs
❡
(f.g)(X) = X
s≥0
csXs, ♦♥❞❡ cs = X
i+j=s
aibj, ❝♦♠ i, j ≥0
❉❛❞♦s ♦s ♣♦❧✐♥ô♠✐♦sf, g, h ∈K[X]✱ ❡st❡s ❣♦③❛♠ ❞❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✐✮ ❈♦♠✉t❛t✐✈✐❞❛❞❡✿ f +g =g+f ❡ f.g =g.f❀
✐✐✮ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ (f+g) +h=f + (g+h) ❡(f.g).h=f.(g.h)❀
✐✐✐✮ ❉✐str✐❜✉t✐✈✐❞❛❞❡✿ f.(g+h) =f.g+f.h✳
❊①❡♠♣❧♦ ✹✳ ❈♦♥s✐❞❡r❡ ♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s f(X) = 1 + 2X + 3X3 ❡
g(X) = X+ 2X3
.
❊♥tã♦
(f +g)(X) = f(X) +g(X) = 1 + 2X+ 3X3
+X+ 2X3
= 1 + 3X+ 5X3
.
❡
(f.g)(X) = (1 + 2X+ 3X3).(X+ 2X3)
= 1.(X+ 2X3) + 2X.(X+ 2X3) + 3X3.(X+ 2X3) = X+ 2X3
+ 2X2
+ 4X4 + 3X4
+ 6X6
)
= X+ 2X2+ 2X3+ 7X4+ 6X6.
❉❡✜♥✐çã♦ ✼✳ ❙❡❥❛ f(X) = a0+a1X+a2X2+· · ·+anXn∈K[X]\ {0}✱ ❝♦♠ an 6= 0✱ ❡am = 0 ♣❛r❛ t♦❞♦ m > n✳ ❆ss✐♠✱ t❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛(a0, a1,· · · , an)é q✉❛s❡ t♦❞❛ ♥✉❧❛✱ ❡♥tã♦ ♦ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦ n é ♦ ❣r❛✉ ❞❡ f✱ ❡ ❞❡♥♦t❛♠♦s ∂f =n.
Pr♦♣♦s✐çã♦ ✷✳ ❙❡❥❛♠ f, g∈K[X]\ {0}✱ t❡♠♦s q✉❡✿
✐✮ ❙❡ f+g 6= 0✱ ❡♥tã♦ ∂(f+g)≤ ♠á①{∂f, ∂g}✳ ✐✐✮ ❙❡ f.g6= 0✱ ❡♥tã♦ ∂(f.g) = ∂f+∂g✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ∂f =n ❡∂g=m✱ ❝♦♠
f(X) =
n X
s=0
asXs ❡ g(X) = m X
s=0
bsXs, as, bs ∈K.
✐✮ ❈♦♥s✐❞❡r❡ ❞♦✐s ❝❛s♦s✿ ✶♦ ❝❛s♦✿ n6=m ❡ n > m.
(f +g)(X) =
n X
s=0
asXs+ m X
s=0
bsXs
=
m X
s=0
asXs+ n X
s=m+1
asXs+ m X
s=0
bsXs
=
m X
s=0
(as+bs)Xs+ n X
s=m+1
asXs
▲♦❣♦ ♦∂(f +g) =n =♠á①{∂f, ∂g}. ✷♦ ❝❛s♦✿ n=m
(f+g)(X) =
n X
s=0
asXs+ m X
s=0
bsXs
=
n X
s=0
asXs+ n X
s=0
bsXs
=
n X
s=0
(as+bs)Xs.
❉❡✈❡♠♦s ✈❡r✐✜❝❛r ❛❣♦r❛ s❡✿ (an+bn) = 0 ♦✉(an+bn)6= 0.
• (an+bn) = 0⇒∂(f +g)< n=♠á①{∂f, ∂g}✳
• (an+bn)6= 0⇒∂(f +g) = n=♠á①{∂f, ∂g}✳
P♦rt❛♥t♦ ❡♠ q✉❛❧q✉❡r ✉♠ ❞♦s ❞♦✐s ❝❛s♦s t❡♠♦s q✉❡✿ ∂(f+g)≤♠á①{∂f, ∂g}.
✐✐✮ P❡❧❛ ❉❡✜♥✐çã♦ ✻✱ t❡♠♦s q✉❡✿
(f.g)(X) = X
s≥0
csXs, ♦♥❞❡ cs= X
i+j=s
aibj, ❝♦♠ i, j ≥0.
❉❛í✱∂(f.g) =s✳ ❈♦♠♦s=i+j✱ ❝♦♠i= 0,1,2,· · · , n ❡ j = 0,1,2,· · · , m✱ t❡♠♦s q✉❡ s=n+m.
❚❡♠♦s q✉❡ anbm 6= 0✱ ♣♦✐s ❛s s❡q✉ê♥❝✐❛✱ (a0, a1,· · · , an) ❡ (b0, b1,· · · , bm)✱ sã♦ q✉❛s❡ t♦❞❛ ♥✉❧❛✱ ❝♦♠ai = 0 ♣❛r❛ i > n ❡ bj = 0✱ ♣❛r❛ j > m.
▲♦❣♦✱
∂(f.g) = s=n+m=∂f +∂g.
❉❡✜♥✐çã♦ ✽✳ ❱❛❧♦r ♥✉♠ér✐❝♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦f(X)∈K[X]é ♦❜t✐❞♦ ♣❡❧❛ s✉❜st✐t✉✐çã♦
❞❛ ✈❛r✐á✈❡❧X ♣♦r ✉♠ ❡❧❡♠❡♥t♦ α ∈K.
❊①❡♠♣❧♦ ✺✳ ❉❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ♥✉♠ér✐❝♦ ❞♦ ♣♦❧✐♥ô♠✐♦ f(X) =X4
−3X3
+ 2X2
−6X+ 1 ♣❛r❛ X = 2.
❖ ✈❛❧♦r ♥✉♠ér✐❝♦ é ♦❜t✐❞♦ s✉❜st✐t✉✐♥❞♦ ❛ ✈❛r✐á✈❡❧X ♣♦r2♥♦ ♣♦❧✐♥ô♠✐♦ ❡ r❡❛❧✐③❛♥❞♦
❛s ♦♣❡r❛çõ❡s ❞❡✈✐❞❛s✱ ❛ss✐♠✱ f(2) = 24
−3(2)3
+ 2(2)2
−6(2) + 1 = 16−24 + 8−12 + 1 =−11✳
P♦rt❛♥t♦✱ f(2) =−11.
▲❡♠❛ ✸ ✭❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦✮✳ ❉❛❞♦s ♦s ♣♦❧✐♥ô♠✐♦s ♥ã♦ ♥✉❧♦s f(X), g(X)∈K[X]✱
❡♥tã♦ ❡①✐st❡♠ ♣♦❧✐♥ô♠✐♦s q(X), r(X)∈K[X]t❛❧ q✉❡ f(X) =q(X).g(X) +r(X), ♦♥❞❡ r(X) = 0 ♦✉ ∂r(X)< ∂g(X).
❉❡♠♦♥str❛çã♦✳ ❆ ♣r♦✈❛ é ♥❛ ✈❡r❞❛❞❡ ♥❛❞❛ ♠❛✐s q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ✧❞✐✈✐sã♦ ❧♦♥❣❛✧ ❛ q✉❛❧ é ✉s❛❞❛ ♥❛ ❡s❝♦❧❛ ♣❛r❛ ❞✐✈✐❞✐r ✉♠ ♣♦❧✐♥ô♠✐♦ ♣♦r ♦✉tr♦✳ ❙❡∂f(X)< ∂g(X) ♥ã♦
❤á ♥❛❞❛ ♣❛r❛ ♣r♦✈❛r✳ ❇❛st❛ ❝♦❧♦❝❛r q(X) = 0✱ ❡ r(X) = f(X)✱ ❡ ❝❡rt❛♠❡♥t❡ t❡r❡♠♦s
f(X) = 0.g(X) +f(X)✱ ♦♥❞❡∂r(X)≤∂g(X)♦✉ f(X) = 0✳
❈♦♥s✐❞❡r❡ ❛❣♦r❛∂f(X)≥∂g(X)✳ ❙❡❥❛♠✱
f(X) =anXn+an−1Xn−1 +· · ·+a2X2+a1X+a0 ❡
g(X) =bmXm+bm−1Xm−1+· · ·+b2X2+b1X+b0, ♦♥❞❡ an 6= 0, bm6= 0 ❡n ≥m. ❚♦♠❛♥❞♦ f1(X) = f(X)−
an bm
Xn−mg(X)✳ ❆ss✐♠✱ ∂f
1(X) ≤ n − 1✱ ❡♥tã♦ ♣♦r
✐♥❞✉çã♦ s♦❜r❡ ♦ ❣r❛✉ ❞❡ f(X)✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ f1(X) = q1(X).g(X) +r(X) ♦♥❞❡
r(X) = 0 ♦✉∂r(X)< ∂g(X)✳
❙✉❜st✐t✉✐♥❞♦f1(X)✱ t❡♠♦s✿
f(X)−
an bm
Xn−mg(X) = q
1(X).g(X) +r(X)⇒
f(X) =
an bm
Xn−mg(X) +q
1(X).g(X) +r(X)⇒
f(X) =
an bm
Xn−m+q 1(X)
.g(X) +r(X).
❈♦♥s✐❞❡r❛♥❞♦✱ q(X) =
an bm
Xn−m+q
1(X)✱ t❡♠♦s
f(X) = q(X).g(X) +r(X), ❝♦♠ q(X), r(X)∈K[X], ♦♥❞❡r(X) = 0 ♦✉ ∂r(X)< ∂g(X).
❊①❡♠♣❧♦ ✻✳ ❖ ♣♦❧✐♥ô♠✐♦ f(X) = X3
+ X2
+ X + 3 ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ (X+ 1)(X2
+ 1) + 2✱ ♦✉ s❡❥❛✱ X3
+X2
+X+ 3 = (X+ 1)(X2
+ 1) + 2. ❙❡♥❞♦ g(X) = X2
+ 1 ❡ q(X) = X+ 1 ❡ r(X) = 2✱ t❡♠♦s q✉❡✿
f(x) =q(X).g(X) +r(X), ❝♦♠ ∂r(X)< ∂g(X).
✷✳✹
❆❧❣♦r✐t♠♦ Prát✐❝♦ ❞❛ ❉✐✈✐sã♦ ❞❡ P♦❧✐♥ô♠✐♦s
P❛r❛ ❞✐✈✐❞✐r ✉♠ ♣♦❧✐♥ô♠✐♦ f(X) = anXn + an−1Xn−1 + · · · + a1X + a0 ♣♦r
g(X) = bmXm + bm−1Xm− 1
+ · · ·+ b1X +b0 ❝♦♠ n ≥ m ❡ an, bm 6= 0✱ ❞❡✈❡♠♦s ❞✐✈✐❞✐r ♦ t❡r♠♦anXn ♣❡❧♦ t❡r♠♦ bmXm ♦❜t❡♥❞♦ ❝♦♠♦ r❡s✉❧t❛❞♦
cn−m = (an.b− 1 m )X
n−m, ❝♦♠ a
n, bm ∈K.
❊♠ s❡❣✉✐❞❛ ♠✉❧t✐♣❧✐❝❛✲s❡cn−m ♣♦r g(X) ❡ s✉❜tr❛✐ ❞❡ f(X) ♦❜t❡♥❞♦ r1(X). ❙❡ ♦ ❣r❛✉
❞❡ r1(X) ❢♦r ♠❡♥♦r q✉❡ ♦ ❣r❛✉ ❞❡ g(X) ❛ ❞✐✈✐sã♦ ❡♥❝❡rr❛✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞❡✈❡♠♦s
r❡♣❡t✐r ♦ ♣r♦❝❡ss♦✱ ✐st♦ é✱ ❞✐✈✐❞✐r ♦ t❡r♠♦ ❞❡ ♠❛✐♦r ♣♦tê♥❝✐❛ ❞❡r1(X)♣❡❧♦ t❡r♠♦bmXm✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦g(X) ❡ s✉❜tr❛✐ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦r1(X)✳
❖ ♣r♦❝❡❞✐♠❡♥t♦ t❡r♠✐♥❛ q✉❛♥❞♦ ♦ ❣r❛✉ ❞❡rs(X) ❢♦r ♠❡♥♦r q✉❡ ♦ ❣r❛✉ ❞❡g(X)✱ ♦♥❞❡ ♦ ♣♦❧✐♥ô♠✐♦ q(X) é ♦❜t✐❞♦ ♣❡❧❛ s♦♠❛tór✐❛ ❞♦s t❡r♠♦s ❞♦ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦✳
❊①❡♠♣❧♦ ✼✳ ❉✐✈✐❞✐r ♦ ♣♦❧✐♥ô♠✐♦ f(X) =X3
+X2
+X+ 3 ♣♦r g(X) = X2
+ 1, ♦♥❞❡ f(X), g(X)∈R[X].
P❡❧♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡s❝r✐t♦ ❛❝✐♠❛ t❡♠♦s q✉❡ c1 =
X3
X2
= X ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡st❡ ✈❛❧♦r ♣❡❧♦ ♣♦❧✐♥ô♠✐♦g(X)❡ s✉❜tr❛✐♥❞♦ ❞❡f(X)♦❜t❡♠♦sr1(X) =X2+ 3✱ ❝♦♠♦ ♦ ❣r❛✉
❞❡r1(X) ❡ ✐❣✉❛❧ ❛♦ ❣r❛✉ ❞❡ g(X) ❞❡✈❡♠♦s ♣r♦ss❡❣✉✐r ❝♦♠ ❛ ❞✐✈✐sã♦✳ ❉✐✈✐❞✐♥❞♦ ❛❣♦r❛
♦ t❡r♠♦ ❞❡ ♠❛✐♦r ❣r❛✉ ❞❡ r1(X) ♣❡❧♦ t❡r♠♦ X2✱ ♦❜t❡♠♦s c0 = 1✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ cs−1
♣♦rg(X) ❡ s✉❜tr❛✐♥❞♦ ❞❡r1(X)✱ ♦❜t❡♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦r2(X) = 2✳ ❈♦♠♦ ♦ ❣r❛✉ ❞❡
r2(X) é ♠❡♥♦r q✉❡ ♦ ❣r❛✉ ❞❡ g(X)✱ ❛ ❞✐✈✐sã♦ ❡stá ❝♦♠♣❧❡t❛✳ P♦rt❛♥t♦✱ t❡♠♦s ❝♦♠♦
r❡s✉❧t❛❞♦✱ r2(X) = 2✱q(X) =X+ 1✳ ❆ss✐♠✱
X3+X2+X+ 3 = (X+ 1)(X2+ 1) + 2.
❖✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ♣♦❞❡ s❡r ❛tr❛✈és ❞♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣rát✐❝♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❞✐✈✐sã♦ ❞❛ ❝❤❛✈❡✭❛♥á❧♦❣♦ ❛♦ ♥ú♠❡r✐❝♦✮✳
X3
+X2
+X+ 3 X2
+ 1
−X3
−X X
X2
+ 3 X+ 1
−X2
−1 2
❆ss✐♠✱
X3
+X2
+X+ 3 = (X+ 1)(X2
+ 1) + 2.
✷✳✺ ❘❛✐③❡s ❞❡ P♦❧✐♥ô♠✐♦s
❉❡✜♥✐çã♦ ✾✳ P❛r❛ f(X) = a0 +a1X +· · · +anXn ∈ K[X]✱ ❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛❞❛ ❛ f é ❛ ❢✉♥çã♦ p:K −→K ❞❛❞❛✱ ♣❛r❛ x∈K✱ ♣♦r
p(x) =anxn+· · ·+a1x+a0✳
◗✉❛♥❞♦ f(X) = c✱ ❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛❞❛ é ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ p(x) = c ♣❛r❛ t♦❞♦ x∈K✳
❉❡✜♥✐çã♦ ✶✵✳ ❈❤❛♠❛♠♦s ❞❡ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ♦✉ ♣♦❧✐♥♦♠✐❛❧ t♦❞❛ ❡q✉❛çã♦ q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ p(x) = 0✱ ♦♥❞❡ p(x) r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛❞❛ ❛
f✱ ❞❛❞♦ ♣♦r
p(x) = anxn+an−1xn− 1
+· · ·+a2x 2
+a1x+a0,
♦♥❞❡ (a0, a1,· · · , an) é ✉♠❛ s❡q✉ê♥❝✐❛ q✉❛s❡ t♦❞❛ ♥✉❧❛✱ ❝♦♠ x∈C✱ t❛❧ q✉❡ an 6= 0✳ ❉❡✜♥✐çã♦ ✶✶✳ ❙❡❥❛ f ∈K[X] ✉♠ ♣♦❧✐♥ô♠✐♦✱ ❝♦♠ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛❞❛
p:K −→K✳ ❯♠ ❡❧❡♠❡♥t♦ α∈K é ✉♠❛ r❛✐③ ❞❡ f s❡ p(α) = 0✳
❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦✳
❊①❡♠♣❧♦ ✽✳ ❖ ♥ú♠❡r♦2 é r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦ X3
−2X2
+X−2✱ ♣♦✐s
p(2) = 23
−2.22
+ 2−2 = 0.
❊①❡♠♣❧♦ ✾✳ ❉❡t❡r♠✐♥❡ ❛s r❛í③❡s ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ p(x) =x2
−4x+ 5. ❉❡✈❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ✈❛❧♦r α✱ t❛❧ q✉❡ p(α) = 0✱ ♦✉ s❡❥❛✱
α2−4α+ 5 = 0. ❙♦♠❛♥❞♦ −1❡♠ ❛♠❜♦s ♦s ❧❛❞♦s t❡♠♦s✿
α2
−4α+ 4 =−1⇒ (α−2)2
=−1⇒α−2 =±√−1⇒
α= 2±√−1⇒α = 2±i✳
❆ss✐♠✱ ❡①✐st❡♠ ❞♦✐s ✈❛❧♦r❡s ♣❛r❛αt❛✐s q✉❡p(α) = 0✱ ♦✉ s❡❥❛✱ ❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧p(x)
♣♦ss✉✐ ❞✉❛s r❛í③❡s ❝♦♠♣❧❡①❛s✳
Pr♦♣♦s✐çã♦ ✸ ✭❚❡st❡ ❞❛ r❛✐③✮✳ ❙❡f ∈K[X]\ {0} ❡ α∈K✱ ❡♥tã♦✿ ✐✳ α é r❛✐③ ❞❡ f s❡✱ ❡ só s❡✱ (X−α)|f(X) ❡♠ K[X]❀
✐✐✳ ❙❡ α ❢♦r r❛✐③ ❞❡ f✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♠❛✐♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♠ t❛❧ q✉❡ (X −α)m ❞✐✈✐❞❡ f(x) ❡♠ K[X]❀
✐✐✐✳ ❙❡ α1,· · · , αs ❢♦r❡♠ r❛í③❡s ❞✉❛s ❛ ❞✉❛s ❞✐st✐♥t❛s ❞❡ f✱ ❡♥tã♦ ♦ ♣♦❧✐♥ô♠✐♦
(X−α1)· · ·(X−αs) ❞✐✈✐❞❡ f(X) ❡♠ K[X]✳
❉❡♠♦♥str❛çã♦✳ ✭⇒✮✐✮ P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ t❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣♦❧✐♥ô♠✐♦s
q[X], r[X]∈K[X]t❛✐s q✉❡
f(X) = (X−α)q(X) +r(X), ❝♦♠ r= 0 ♦✉ ∂r < ∂(X−α) = 1. ❆ss✐♠✱ r(X) =c✱ ✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡✳ ❉❛í✱
f(X) = (X−α)q(X) +c.
❙❡♥❞♦p(x) ❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛❞❛ ❛♦ ♣♦❧✐♥ô♠✐♦f(X)✱ t❡♠♦s q✉❡
p(x) = (x−α)q(x) +c. ❊✱ s❡♥❞♦α r❛✐③ ❞❡ f✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✶✶✱ t❡♠♦s
p(α) = (α−α)q(α) +c= 0 =⇒c= 0. P♦rt❛♥t♦✱
f(X) = (X−α)q(X). ▲♦❣♦✱ (X−α)|f(x)✳
✭⇐✮ ❈♦♥s✐❞❡r❡ (X−α)|f(X)✳ ❆ss✐♠ f(X) ♣♦❞❡rá s❡r ❡s❝r✐t♦ ❝♦♠♦✿
f(X) = (X−α)g(X).
❙❡♥❞♦p(x) ❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❛ss♦❝✐❛❞❛ ❛f(X)✱ t❡♠♦s q✉❡✿
p(x) = (x−α)g(x). ❈❛❧❝✉❧❛♥❞♦p(α)✱ ❝♦♠ α∈K✱ ♦❜t❡♠♦s
p(α) = (α−α)g(α) = 0.
