➜➵✐ ❍ä❝ ❚❤➳✐ ◆❣✉②➟♥ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❑❤♦❛ ❍ä❝ ❍♦➭♥❣ ý ỗ ỹ từ ì tứ s ❈❤✉②➟♥ ♥❣➭♥❤ ✿ P❤➢➡♥❣ P❤➳♣ ❚♦➳♥ ❙➡ ❈✃♣ ▼➲ sè✿ ✻✵✳✹✻✳✹✵ ▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙Ü ❚♦➳♥ ❍ä❝ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ P●❙✳❚❙✳ ➜➭♠ ❱➝♥ ◆❤Ø ❚❤➳✐ ◆❣✉②➟♥ ✲ ✷✵✶✶ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ❈➠♥❣ tr×♥❤ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❑❤♦❛ ❍ä❝ ✲ ➜➵✐ ❍ä❝ ❚❤➳✐ ◆❣✉②➟♥ P❤➯♥ ❜✐Ö♥ ✶✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ P❤➯♥ ❜✐Ö♥ ✷✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ▲✉❐♥ ✈➝♥ sÏ ➤➢ỵ❝ ❜➯♦ ✈Ư tr➢í❝ ❤é✐ ➤å♥❣ ❝❤✃♠ ❧✉❐♥ ✈➝♥ ❤ä♣ t➵✐✿ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❑❤♦❛ ❍ä❝ ✲ ➜➵✐ ❍ä❝ ❚❤➳✐ ◆❣✉②➟♥ ◆❣➭②✳✳✳✳ t❤➳♥❣✳✳✳✳ ♥➝♠ ✷✵✶✶ ❈ã t❤Ĩ t×♠ ❤✐Ĩ✉ t➵✐ ❚❤➢ ❱✐Ư♥ ➜➵✐ ❍ä❝ ❚❤➳✐ ◆❣✉②➟♥ S hóa b i Trung tâm H c li u – i h c Thái Ngun DeThiMau.vn http://www.lrc-tnu.edu.vn ▼ơ❝ ❧ơ❝ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✳✶ ✶✳✷ ✹ ❑❤➳✐ ♥✐Ö♠ ✈➭♥❤ ✈➭ ➤å♥❣ ❝✃✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✶ ❱➭♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ➛í❝ ❝đ❛ ❦❤➠♥❣✳ ▼✐Ị♥ ♥❣✉②➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✸ ➜å♥❣ ❝✃✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✹ ❚r➢ê♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❱➭♥❤ ➤❛ t❤ø❝ ✈➭ ♥❣❤✐Ö♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✳ ✳ ✳ ✳ ✳ ỗ ũ từ ì tứ ỗ ũ từ ❤×♥❤ t❤ø❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷ ❉➲② ❤✐Ư✉ ❝đ❛ ♠ét ❞➲② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸ ❍➭♠ s✐♥❤ t❤➢ê♥❣ ✈➭ ❞➲② ❋✐❜♦♥❛❝❝✐✱ ❞➲② ❈❛t❛❧❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹ ❍➭♠ s✐♥❤ ♠ị ✈➭ ❞➲② sè ❙t✐r❧✐♥❣ ✷✳✺ ❍➭♠ s✐♥❤ ❝đ❛ ❞➲② ❝➳❝ ➤❛ t❤ø❝ ❇❡r♥♦✉❧❧✐ ✷✳✻ ❍➭♠ s✐♥❤ ❉✐r✐❝❤❧❡t ✈➭ ❤➭♠ ❩❡t❛✲❘✐❡♠❛♥♥ ✷✳✼ ❚Ý❝❤ ✈➠ ❤➵♥ ✳ ✷✳✽ ➜å♥❣ ♥❤✃t t❤ø❝ ◆❡✇t♦♥ ✷✳✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❉➲② tr✉② ❤å✐ ✈í✐ ❤➭♠ s✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✶ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ▼ë ➤➬✉ ❚r♦♥❣ t♦➳♥ ❤ä❝ ✈✐Ư❝ sư ❞ơ♥❣ ❝➳❝ ❦✐Õ♥ t❤ø❝ t♦➳♥ ❝❛♦ ❝✃♣ ➤Ó ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ë ♣❤ỉ t❤➠♥❣ ❧➭ ➤✐Ị✉ r✃t q✉❛♥ trä♥❣✳ ◆ã ❦❤➠♥❣ ❝❤Ø ❣✐ó♣ ♥❣➢ê✐ ❧➭♠ t♦➳♥ ❝ã ♥❤✐Ị✉ ♣❤➢➡♥❣ ♣❤➳♣ ❧ù❛ ❝❤ä♥ ❧ê✐ ❣✐➯✐✱ ♠ë ré♥❣ t➬♠ ❤✐Ó✉ ❜✐Õt t♦➳♥ ❤ä❝ ♠➭ ò t ợ t sứ s t➵♦✱ t➬♠ ❜❛♦ q✉➳t ❜➭✐ t♦➳♥✱ ♠ë ré♥❣ ❜➭✐ t♦➳♥ ❞➢í✐ ♥❤✐Ị✉ ❤➢í♥❣ ❦❤➳❝ ♥❤❛✉✳ ❙ư ❞ơ♥❣ ❝➳❝ ❦✐Õ♥ t❤ø❝ ề ỗ số ể qết t ề ❞➲② sè ❧➭ ♠ét ✈✃♥ ➤Ị ♥❤➢ ✈❐②✳ ◆❤➢ ❝❤ó♥❣ t❛ ➤➲ ❜✐Õt ❝➳❝ ✈✃♥ ➤Ò ❧✐➟♥ q✉❛♥ ➤Õ♥ ❞➲② sè ❧➭ ♠ét ♣❤➬♥ q✉❛♥ trä♥❣ ❝ñ❛ ➤➵✐ sè ✈➭ ❣✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝✳ ❑❤✐ t✐Õ♣ ❝❐♥ ✈✃♥ ➤Ò ♥➭② ❝➳❝ ❡♠ ❤ä❝ s✐♥❤ ❣✐á✐✱ s✐♥❤ ✈✐➟♥ ✈➭ ❦❤➳ ♥❤✐Ò✉ t❤➬② ❝➠ ❣✐➳♦ ♣❤æ t❤➠♥❣ t❤➢ê♥❣ r✃t ♣❤➯✐ ➤è✐ ♠➷t ✈í✐ r✃t ♥❤✐Ị✉ ❜➭✐ t♦➳♥ ❦❤ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ❝❤✉②➟♥ ➤Ò ♥➭②✳ ❚r♦♥❣ ❝➳❝ ❦ú t❤✐ ❤ä❝ s✐♥❤ ❣✐á✐ q✉è❝ ❣✐❛✱ t❤✐ ❖❧✐♠♣✐❝ t♦➳♥ q✉è❝ tÕ✱ t❤✐ ❖❧✐♠♣✐❝ t♦➳♥ s✐♥❤ ✈✐➟♥ ❣✐÷❛ ❝➳❝ tr➢ê♥❣ ➤➵✐ ❤ä❝✱ ❝❛♦ ➤➻♥❣✱ ❝➳❝ ❜➭✐ t♦➳♥ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❞➲② sè ❝ị♥❣ ❤❛② ➤➢ỵ❝ ➤Ị ❝❐♣ ✈➭ t❤➢ê♥❣ ❧♦➵✐ r✃t ❦❤ã✱ ➤ß✐ ❤á✐ ♥❣➢ê✐ ❤ä❝✱ ♥❣➢ê✐ ❧➭♠ t♦➳♥ ♣❤➯✐ ❝ã ♠ét t➬♠ ❤✐Ó✉ ❜✐Õt ré♥❣ ✈➭ r✃t s➞✉ s➽❝ ❝➳❝ ❦✐Õ♥ t❤ø❝ ✈Ò ❞➲② sè ỗ số r t♦➳♥ ❤❛② ✈➭ ❤♦➭♥ t❤✐Ư♥ ➤➢ỵ❝ ❜➭✐ t♦➳♥✳ ➜Ĩ ♣❤ơ❝ ✈ơ ❝❤♦ ✈✐Ư❝ ❜å✐ ❞➢ì♥❣ ❤ä❝ s✐♥❤ ❣✐á✐ ✈➭ ✈✐Ư❝ tr❛♦ ➤ỉ✐ ❦✐♥❤ ♥❣❤✐Ư♠ ✈í✐ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ ❜å✐ ❞➢ì♥❣ ❤ä❝ s✐♥❤ ❣✐á✐ q✉❛♥ t➞♠ ✈➭ t×♠ ❤✐Ĩ✉ t❤➟♠ ✈Ị ♣❤➬♥ ♥➭②✱ ➤➢ỵ❝ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ t❤➬② ➜➭♠ ❱➝♥ ◆❤Ø t➳❝ ❣✐➯ ➤➲ ❤ä❝ t❐♣ t❤➟♠ ✈➭ ✈✐Õt ề t ỗ ỹ từ ì tứ s✐♥❤✧✳ ➜Ò t➭✐ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ò trä♥❣ t➞♠ ✿ ❈❤➢➡♥❣ ■ ✿ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳❚➳❝ ❣✐➯ ♥❤➽❝ ❧➵✐ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ♥❤✃t ✈Ò ✿ ✶✳✶ ❑❤➳✐ ♥✐Ö♠ ✈➭♥❤ ✈➭ ➤å♥❣ ❝✃✉ ✶✳✶✳✶ ❱➭♥❤✳ ✶✳✶✳✷ ➛í❝ ❝đ❛ ❦❤➠♥❣✳ ▼✐Ị♥ ♥❣✉②➟♥✳ ✷ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✸ ✶✳✶✳✸ ➜å♥❣ ❝✃✉✳ ✶✳✶✳✹ ❚r➢ê♥❣✳ ✶✳✷ ❱➭♥❤ ➤❛ t❤ø❝ ✈➭ ♥❣❤✐Ö♠✳ ❈❤➢➡♥❣ ■■ ✿ ❱➭♥❤ ỗ ỹ từ ì tứ tệ ế tứ ỗ ỹ từ ì t❤ø❝✳ ✷✳✷ ❉➲② ❤✐Ư✉ ❝đ❛ ♠ét ❞➲② ✳ ✷✳✸ ❍➭♠ s✐♥❤ t❤➢ê♥❣ ✈➭ ❞➲② ❋✐❜♦♥❛❝❝✐✱ ❞➲② ❈❛t❛❧❛♥✳ ✷✳✹ ❍➭♠ s✐♥❤ ♠ị ✈➭ ❞➲② sè ❙t✐r❧✐♥❣✳ ✷✳✺ ❍➭♠ s✐♥❤ ❝đ❛ ❞➲② ❝➳❝ ➤❛ t❤ø❝ ❇❡r♥♦✉❧❧✐✳ ✷✳✻ ❍➭♠ s✐♥❤ ❉✐r✐❝❤❧❡t ✈➭ ❤➭♠ ❩❡t❛✲❘✐❡♠❛♥♥✳ ✷✳✼ ❚Ý❝❤ ✈➠ ❤➵♥✳ ✷✳✽ ➜å♥❣ ♥❤✃t t❤ø❝ ◆❡✇t♦♥✳ ✷✳✾ ❉➲② tr✉② ❤å✐ ✈í✐ ❤➭♠ s✐♥❤✳ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ✈➭ ❝❤Ø ❜➯♦ t❐♥ t×♥❤ ❝đ❛ P●❙✳❚❙ ➜➭♠ ❱➝♥ ◆❤Ø ✲ ➜➵✐ ❤ä❝ ❙➢ P❤➵♠ ❍➭ ◆é✐✳ ❚❤➬② ➤➲ ❞➭♥❤ ♥❤✐Ò✉ t❤ê✐ ❣✐❛♥ ❤➢í♥❣ ❞➱♥ ✈➭ ❣✐➯✐ ➤➳♣ ❝➳❝ t❤➽❝ ♠➽❝ ❝đ❛ t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❧➭♠ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ➤Õ♥ ❚❤➬②✳ ❚➳❝ ❣✐➯ ①✐♥ ❣ư✐ tí✐ ❝➳❝ t❤➬② ✭❝➠✮ ❦❤♦❛ ❚♦➳♥✱ ♣❤ß♥❣ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❑❤♦❛ ❍ä❝ ✲ ➜➵✐ ❍ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ❝ï♥❣ ❝➳❝ t❤➬② ❝➠ t❤❛♠ ❣✐❛ ❣✐➯♥❣ ❞➵② ❦❤ã❛ ❈❛♦ ❤ä❝ ✷✵✵✾✲✷✵✶✶ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ ề ỗ tr tờ q t❤ê✐ ①✐♥ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ t❐♣ t❤Ĩ ❧í♣ ❈❛♦ ❤ä❝ ❚♦➳♥ ❑✸❇ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❑❤♦❛ ❍ä❝ ➤➲ ➤é♥❣ ✈✐➟♥ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚➳❝ ❣✐➯ ①✐♥ ❝➯♠ ➡♥ tí✐ ❙ë ◆é✐ ❱ơ✱ ❙ë ●✐➳♦ ❞ơ❝ ✈➭ ➤➭♦ t➵♦ ❇➽❝ ◆✐♥❤✱ ❇❛♥ ❣✐➳♠ ❤✐Ư✉ ✈➭ tỉ ❚♦➳♥ tr➢ê♥❣ ❚❍P❚ ▲➢➡♥❣ ❚➭✐ ✷ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❣✐ó♣ ➤ì ➤Ĩ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ❦❤ã❛ ❤ä❝ ♥➭②✳ ❚➳❝ ❣✐➯ ❍♦➭♥❣ ❱➝♥ ◗✉ý S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ❈❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✳✶ ✶✳✶✳✶ ❑❤➳✐ ♥✐Ư♠ ✈➭♥❤ ✈➭ ➤å♥❣ ❝✃✉ ❱➭♥❤ ➜Þ♥❤ ♥❣❤Ü❛ ✳ ❚❛ ❣ä✐ ❧➭ ✈➭♥❤ ♠ét t❐♣ ❤ỵ♣ ❳ ❝ï♥❣ ✈í✐ ❤❛✐ ♣❤Ð♣ t♦➳♥ ❤❛✐ ♥❣➠✐ ➤➲ ❝❤♦ tr♦♥❣ ❳ ❦ý ❤✐Ö✉ t❤❡♦ t❤ø tù ❜➺♥❣ ❝➳❝ ❞✃✉ ✰ ✈➭ ✳ ✭♥❣➢ê✐ ❤✐Ö✉ ♥❤➢ ✈❐②✮ t❛ t❤➢ê♥❣ ❦ý ✈➭ ❣ä✐ ❧➭ ♣❤Ð♣ ❝é♥❣ ✈➭ ♣❤Ð♣ ♥❤➞♥ s❛♦ ❝❤♦ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥✿ ✶✮ ❳ ❝ï♥❣ ✈í✐ ♣❤Ð♣ ❝é♥❣ ❧➭ ♠ét ♥❤ã♠ ❛❜❡♥✳ ✷✮ ❳ ❝ï♥❣ ✈í✐ ♣❤Ð♣ ♥❤➞♥ ❧➭ ♠ét ♥ư❛ ♥❤ã♠✳ ✸✮ P❤Ð♣ ♥❤➞♥ ♣❤➞♥ ♣❤è✐ ✈í✐ ♣❤Ð♣ ❝é♥❣✿ ❱í✐ ❝➳❝ ♣❤➬♥ tö tï② ý ❝ã✿ x, y, z ∈ X t❛ x(y + z) = xy + xz (y + z)x = yx + zx P❤➬♥ tö tr✉♥❣ ❧❐♣ ❝đ❛ ♣❤Ð♣ ❝é♥❣ t❤× ❦ý ❤✐Ư✉ ❧➭ ✵ ✈➭ ❣ä✐ ❧➭ ♣❤➬♥ tư ❦❤➠♥❣✳ P❤➬♥ tư ➤è✐ ①ø♥❣ ✭➤è✐ ✈í✐ ♣❤Ð♣ ❝é♥❣ ✮ ❝đ❛ ♠ét ♣❤➬♥ tư ① t❤× ❦ý ❤✐Ö✉ ❧➭ ✲① ✈➭♥❤ ❳ ❧➭ ❣✐❛♦ ❤♦➳♥✳ ◆Õ✉ ♣❤Ð♣ ♥❤➞♥ ❝ã ♣❤➬♥ tư tr✉♥❣ ❧❐♣ t❤× ♣❤➬♥ tư ➤ã ❣ä✐ ❧➭ ♣❤➬♥ tư ➤➡♥ ✈Þ ❝đ❛ ① ✈➭ t❤➢ê♥❣ ❦Ý ❤✐Ö✉ ❧➭ ❡ ❤❛② ✶ ✳ ✈➭ ❣ä✐ ❧➭ ➤è✐ ❝ñ❛ ① ✳ ◆Õ✉ ♣❤Ð♣ ♥❤➞♥ ❧➭ ❣✐❛♦ ❤♦➳♥ tì t ủ ề ị ♥❣❤Ü❛✶ ✿ ❚❛ ❣ä✐ ❧➭ ♠➲♥ q✉❛♥ ❤Ư ❛❜❂✵✳ ➜Þ♥❤ ♥❣❤Ü❛✷ ✿ ❚❛ ❣ä✐ ➢í❝ ❝đ❛ ✵ ♠ä✐ ♣❤➬♥ tư a = s❛♦ ❝❤♦ ❝ã b = t❤á❛ ♠✐Ị♥ ♥❣✉②➟♥ ♠ét ✈➭♥❤ ❝ã ♥❤✐Ị✉ ❤➡♥ ♠ét ♣❤➬♥ tư✱ ❣✐❛♦ ✹ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✺ ❤♦➳♥✱ ❝ã ➤➡♥ ✈Þ✱ ❦❤➠♥❣ ❝ã ➢í❝ ❝đ❛ ✶✳✶✳✸ ✵✳ ➜å♥❣ ❝✃✉ ➜Þ♥❤ ♥❣❤Ü❛✳ ▼ét ➤å♥❣ ❝✃✉ ✭✈➭♥❤✮ ❧➭ ♠ét ➳♥❤ ①➵ tõ ♠ét ✈➭♥❤ ❳ ➤Õ♥ ♠ét ✈➭♥❤ ❨ s❛♦ ❝❤♦✿ f (a + b) = f (a) + f (b) f (ab) = f (a) f (b) ✈í✐ ♠ä✐ a, b ∈ X ◆Õ✉ X = Y t❤× ➤å♥❣ ❝✃✉ f ❣ä✐ ❧➭ ♠ét tù ➤å♥❣ ❝✃✉ ❝đ❛ X✳ ❚❛ ❝ị♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ➤➡♥ ❝✃✉✱ t♦➭♥ ❝✃✉✱ ➤➻♥❣ ❝✃✉ t➢➡♥❣ tù ♥❤➢ ➤➲ ➤Þ♥❤ ♥❣❤Ü❛ tr♦♥❣ ♥❤ã♠✳ ✶✳✶✳✹ ❚r➢ê♥❣ ➜Þ♥❤ ♥❣❤Ü❛✿ ❚❛ ❣ä✐ ❧➭ tr➢ê♥❣ ♠ét ♠✐Ị♥ ♥❣✉②➟♥ ❳ tr♦♥❣ ➤ã ♠ä✐ ♣❤➬♥ tư ❦❤➳❝ ❦❤➠♥❣ ➤Ị✉ ❝ã ♠ét ♥❣❤Þ❝❤ ➤➯♦ tr♦♥❣ ✈Þ ♥❤ã♠ ♥❤➞♥ ❳✳ ❱❐② ♠ét ✈➭♥❤ ❳ ❣✐❛♦ ❤♦➳♥✱ ❝ã ➤➡♥ ✈Þ✱ ❝ã ♥❤✐Ị✉ ❤➡♥ ♠ét ♣❤➬♥ tö ❧➭ ♠ét tr➢ê♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ X − {0} ❧➭ ♠ét ♥❤ã♠ ➤è✐ ✈í✐ ♣❤Ð♣ ♥❤➞♥ ❝đ❛ ❳✳ ✶✳✷ ❱➭♥❤ ➤❛ t❤ø❝ ✈➭ ♥❣❤✐Ư♠ ❑Õt q✉➯ ❝❤Ý♥❤ ❈❤♦ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ R ✈➭ ♠ét ❜✐Õ♥ x tr➟♥ R ❱í✐ ❝➳❝ n ∈ N, ①Ðt t❐♣ ❤ỵ♣✿ n n R[x] = {a0 + a1 x + a2 x + · · · + an x | ∈ R} = i=0 xi | ∈ R f (x) ∈ R[x] ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ➤❛ t❤ø❝ ❝đ❛ ❜✐Õ♥ x ✈í✐ ❝➳❝ ❤Ư sè t❤✉é❝ ✈➭♥❤ R ❍Ư sè an ➤➢ỵ❝ ❣ä✐ ❧➭ ệ số t ò ệ số a0 ợ ọ ❧➭ ❤Ư sè tù ❞♦ ❝đ❛ f (x) ❑❤✐ an = tì n ợ ọ ủ f (x) ợ ỗ tử S húa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✻ ❦ý ❤✐Ö✉ deg f (x) tứ ợ q ị ó m i x , g(x) = bi xi ∈ R[x] t❤× i=0 i=0 n ◆Õ✉ f (x) = f (x) = g(x) ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ m = n, = bi ✈í✐ ♠ä✐ −∞ ❤♦➷❝ i −1 n i i f (x) + g(x) = (ai + bi )x , f (x)g(x) = i=0 ➜Þ♥❤ ❧ý ✶✳✷✳✶✳ ❚❛ ❝ã ♠✐Ị♥ ♥❣✉②➟♥ t❤× R[x] ai−j bj )xi ( i=0 j=0 ❧➭ ♠ét ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ R ❧➭ ♠ét R[x] ❝ị♥❣ ❧➭ ♠ét ♠✐Ị♥ ♥❣✉②➟♥✳ f (x), g(x) ∈ k[x] ✈➭ g(x) = ❝ã ❤❛✐ ➤❛ t❤ø❝ ❞✉② ♥❤✃t q(x), r(x) s❛♦ ❝❤♦ f (x) = q(x)g(x) + r(x) ✈í✐ deg r(x) < deg g(x) ➜Þ♥❤ ❧ý ✶✳✷✳✷✳ ●✐➯ sư k ❧➭ ♠ét tr➢ê♥❣✳ ❱í✐ ❝➳❝ ➤❛ t❤ø❝ n ✈➭ p ✈í✐ n > p p ✈➭ ➤đ ➤Ĩ x − a ❝❤✐❛ ❤Õt ❝❤♦ x − a ✈í✐ a ∈ R, a = ❱Ý ❞ô ✶✳✷✳✸✳ ❈❤♦ ❤❛✐ sè tù ♥❤✐➟♥ n ❇➭✐ ❣✐➯✐✿ n ❚×♠ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ p ❇✐Ĩ✉ ❞✐Ơ♥ n = qp + r Z ✈í✐ tr♦♥❣ r < p ❑❤✐ ➤ã ❝ã ❜✐Ĩ✉ ❞✐Ơ♥ xn − an = (xp − ap )(xn−p + ap xn−2p + · · · + a(q−1)p xn−qp ) + aqp (xr − ar ) xn − an ❱❐②✱ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤Ĩ ➜Þ♥❤ ❧ý ✶✳✷✳✹✳ ●✐➯ sư k ❝❤✐❛ ❤Õt ❝❤♦ xp − ap ❧➭ ♠ét tr➢ê♥❣✳ ❑❤✐ ➤ã ✈➭♥❤ k[x] ❧➭ n :˙ p ❧➭ ♠ét ✈➭♥❤ ❝❤Ý♥❤ ✈➭ ♥ã ❧➭ ✈➭♥❤ ♥❤➞♥ tö ❤ã❛✳ n α ∈ R ●✐➯ sö n α i ∈ R i=0 ❣ä✐ ❧➭ ♠ét ✈➭ ➤❛ t❤ø❝ f (x) = i=0 ợ ọ trị f (x) t ế ❇✐Ĩ✉ t❤ø❝ f (α) = t❤× ✈➭ f (α) = α ➤➢ỵ❝ α ∈ k f (α) = ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♥❣❤✐Ư♠ ❜é✐ ❝✃♣ m ❝đ❛ f (x) tr♦♥❣ k ♥Õ✉ f (x) ❝❤✐❛ m m+1 ❤Õt ❝❤♦ (x − α) ✈➭ f (x) ❦❤➠♥❣ ❝❤✐❛ ❤Õt ❝❤♦ (x − α) ♥❣❤✐Ư♠ ❝đ❛ ➜Þ♥❤ ❧ý ✶✳✷✳✺✳ ➜❛ t❤ø❝ ✭✐✮ ✭✐✐✮ ◆Õ✉ f (x) ❝ñ❛ xi ∈ R[x] tr♦♥❣ R ●✐➯ sö sè ♥❣✉②➟♥ f (x) ∈ k[x] ❜❐❝ n m ❑❤✐ ➤ã t❛ ❝ã ❝➳❝ ❦Õt q✉➯ s❛✉✿ α ∈ k ❧➭ ♥❣❤✐Ö♠ ❝đ❛ f (x) t❤× f (x) = (x − α)g(x) ✈í✐ g(x) ∈ k[x] f (x) ❝ã ❦❤➠♥❣ q✉➳ n ♥❣❤✐Ư♠ ♣❤➞♥ ❜✐Ưt tr♦♥❣ k S hóa b i Trung tâm H c li u – i h c Thái Ngun DeThiMau.vn http://www.lrc-tnu.edu.vn ✼ ➜➠✐ ❦❤✐ ➤Ĩ t×♠ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝➳❝ ♥❣❤✐Ư♠ ❤❛② ♠ét tÝ♥❤ ❝❤✃t ♥➭♦ ➤ã ❝đ❛ ♥❣❤✐Ư♠ ➤❛ t❤ø❝ t❛ t❤➢ê♥❣ sư ❞ơ♥❣ ❦Õt q✉➯ s❛✉ ➤➞②✿ x1 , , xn ❧➭ n ♥❣❤✐Ư♠ ❝đ❛ ➤❛ t❤ø❝ ❜❐❝ n s❛✉ + δ2 xn−2 − · · · + (−1)n δn ❑❤✐ ➤ã ❝ã ❝➳❝ ❤Ư t❤ø❝ ➜Þ♥❤ ❧ý ✶✳✷✳✻✳ ❬❱✐Ðt❪ ●✐➯ sö ➤➞②✿ n n−1 f (x) = x − δ1 x δ = x1 + x2 + · · · + xn δ = x x + x x + · · · + x x 2 n−1 n δ = x x x n n f (x1 , x2 , , xn ) ∈ k[x1 , x2 , , xn ] ❧➭ ♠ét ➤❛ t❤ø❝ ➤è✐ ①ø♥❣ ❦❤➳❝ ✵✳ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét ✈➭ ❝❤Ø ♠ét ➤❛ t❤ø❝ s(x1 , x2 , , xn ) ∈ k[x1 , x2 , , xn ] s❛♦ ❝❤♦ f (x1 , x2 , , xn ) = s(δ1 , δ2 , , δn ) ➜Þ♥❤ ❧ý ✶✳✷✳✼✳ ●✐➯ sư ▼ét sè ✈Ý ❞ô √ f (x) = x4 − 5x3 + 9x2 − 10x + 28 ❚Ý♥❤ f (1 + 3) √ 3 ❧➭ ♥❣❤✐Ư♠ ❝đ❛√g(x) = x3 − 3x2 + 3x − = ✈➭ ❇➭✐ ❣✐➯✐✿ ❱× + f (x) = (x − 2)g(x) + 20 ♥➟♥ f (1 + 3) = 20 ❱Ý ❞ô ✶✳✷✳✽✳ ●✐➯ sö f (x) = a0 xn +a1 xn−1 +· · ·+an−1 x+an ∈ R[x] ✈í✐ a0 = ✈➭ t❤á❛ ♠➲♥ f (x)f (2x2 ) = f (2x3 + x) ✈í✐ ♠ä✐ ❣✐➳ trÞ t❤ù❝ x ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ f (x) ❦❤➠♥❣ t❤Ĩ ❝ã ♥❣❤✐Ư♠ t❤ù❝✳ ❱Ý ❞ơ ✶✳✷✳✾✳ ❬❱▼❖ ✶✾✾✵❪ ●✐➯ sö x3n ✈➭ x0 ë ❤❛✐ ✈Õ✱ ♥➟♥ tõ f (x)f (2x2 ) = f (2x3 + x) t❛ s✉② r❛ a20 = a0 ✈➭ a2n = an ❱× a0 = ♥➟♥ a0 = 1; ❝ß♥ an = ❤♦➷❝ an = ◆Õ✉ an = t❤× f (x) = xr g(x) ✈í✐ g(0) = ❱❐② xr g(x)2r x2r g(2x2 ) = xr (2x2 + 1)r g(2x3 + x) ❤❛② g(x)2r x2r g(2x2 ) = (2x2 + 1)r g(2x3 + x) ❱× g(0) = ♥➟♥ t❛ ♥❤❐♥ ➤➢ỵ❝ g(0) = : ♠➞✉ t❤✉➱♥✳ ❱❐② an = ●✐➯ sö f (x) = ❝ã ♥❣❤✐Ö♠ t❤ù❝ x0 ❑❤✐ ➤ã x0 = ✈× an = ❱× f (2x30 + x0 ) = f (x0 )f (2x20 ) = ♥➟♥ x1 = 2x30 + x0 ❝ị♥❣ ❧➭ ♥❣❤✐Ư♠ t❤ù❝ ❝đ❛ f (x) ❱× ❤➭♠ y = 2x3 + x ❧➭ ➤➡♥ ➤✐Ö✉ t➝♥❣ ♥➟♥ ❞➲② (xr+1 = 2x3r + xr )r ✈➭ x0 = ❧➭ ♠ét ỗ số ề ệ ❝đ❛ f (x) ❤❛② f (x) ❝ã ♥❤✐Ị✉ ✈➠ ❤➵♥ ♥❣❤✐Ư♠✿ ♠➞✉ t❤✉➱♥ t❤❡♦ ➜Þ♥❤ ❧ý ✶✳✷✳✺✳ ❱❐② f (x) ❦❤➠♥❣ ❝ã ♥❣❤✐Ư♠ ❇➭✐ ❣✐➯✐✿ ❙♦ s➳♥❤ ❤Ư sè ❝đ❛ t❤ù❝✳ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✽ a ∈ (0; 1) tr×♥❤ cos 3πa + cos 2πa = ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ a = ❱Ý ❞ơ ✶✳✷✳✶✵✳ ❬■▼❖ ✶✾✾✶❪ ●✐➯ sư sè ❤÷✉ tû t❤á❛ ♠➲♥ ♣❤➢➡♥❣ x = cos πa ❑❤✐ ➤ã 4x3 + 4x2 − 3x − = ❤❛② (2x + 1)(2x2 + −1 −1 t❤× a = ◆Õ✉ x = t❤× 2x + x − = x − 2) = ◆Õ✉ cos πa = x = √ −1 + 17 ❇➺♥❣ 0, ✈➭ ♥❤➢ ✈❐② x ❧➭ sè ✈➠ tû✳ ❉♦ |x| ♥➟♥ cos πa = x = √ an + bn 17 n q✉② ♥➵♣✱ ❝ã t❤Ĩ ❝❤Ø r❛ cos πa = ✈í✐ sè ♥❣✉②➟♥ ❧❰ an , bn ❱× √ √ an+1 + bn+1 17 a + b n n 17 = cos 2n+1 πa = cos2 2n πa − = 2[ ] −1 4 ❇➭✐ ❣✐➯✐✿ ➜➷t a2n + 17b2n − ♥➟♥ an+1 = > an ❉♦ ➤ã ❞➲② (an ) ❧➭ ♠ét ❞➲② t➝♥❣ ♥❣❤✐➟♠ n ♥❣➷t ✈➭ ♥❤➢ ✈❐② t❐♣ ❝➳❝ ❣✐➳ trÞ ❝đ❛ cos πa ✈í✐ n = 0, 1, 2, ❧➭ t❐♣ ✈➠ √ ❤➵♥ ✭✯✮ ✈× 17 ❧➭ sè ✈➠ tû✳ ◆❤➢♥❣ ❞♦ a ❧➭ sè ❤÷✉ tû ♥➟♥ t❐♣ ❝➳❝ ❣✐➳ trÞ ❝đ❛ cos mπa ✈í✐ m = 0, 1, 2, ♣❤➯✐ ❧➭ ❤÷✉ ❤➵♥✿ ♠➞✉ t❤✉➱♥ ✈í✐ ✭✯✮✳ ❉♦ ❞ã a= f (x) ❜❐❝ n ❝ã t✃t ❝➯ ❝➳❝ ♥❣❤✐Ư♠ ➤Ị✉ ′ ❑❤✐ ➤ã t✃t ❝➯ ❝➳❝ ♥❣❤✐Ư♠ ❝đ❛ af (x) + f (x) ❝ị♥❣ ❧➭ ♥❤÷♥❣ sè t❤ù❝✳ ❱Ý ❞ơ ✶✳✷✳✶✶✳ ●✐➯ t❤✐Õt ➤❛ t❤ø❝ f (x) x , x2 , , x k r1 , r2 , , rk ✈➭ t❛ s➽♣ ①Õ♣ x1 < x2 < · · · < xk ❍➭♠ sè ❇➭✐ ❣✐➯✐✿ ●✐➯ sư ❝ã ❝➳❝ ♥❣❤✐Ư♠ t❤ù❝ t❤ù❝✳ ✈í✐ ❜é✐ t➢➡♥❣ ø♥❣ 1 f ′ (x) + + ··· + = g(x) = f (x) x − x1 x − x2 x − xk ❧➭ ❤➭♠ ❧✐➟♥ tô❝ tr♦♥❣ ❝➳❝ ❦❤♦➯♥❣ (−∞; x1 ), (x1 ; x2 ), , (xk−1 ; xk ), (xk ; ∞) ❉ù❛ ✈➭♦ sù ❜✐Õ♥ t❤✐➟♥ ❝đ❛ ❝➳❝ ❤➭♠ , ♣❤➢➡♥❣ tr×♥❤ g(x) = −a ❝ã t❤➟♠ x − xj k ♥❣❤✐Ư♠ ♠í✐ ♥÷❛ ❦❤➳❝ x1 , x2 , , xk ❦❤✐ a = ❱❐② f (x)[g(x) + a] = ❝ã t✃t ❝➯ (r1 − 1) + · · · + (rk − 1) + k = deg f (x) ♥❣❤✐Ư♠ t❤ù❝✳ ❱❐② t✃t ❝➯ ❝➳❝ ′ ♥❣❤✐Ư♠ ❝đ❛ af (x) + f (x) ➤Ò✉ t❤ù❝✳ ❑❤✐ a = t❤× g(x) = ❝ã k − ♥❣❤✐Ư♠ t❤ù❝ ♠í✐ ♥÷❛✳ ❱❐② f (x)[g(x)+0] = ❝ã t✃t ❝➯ (r1 −1)+· · ·+(rk −1)+k−1 = deg f ′ (x) ❚ã♠ ❧➵✐ t✃t ❝➯ ❝➳❝ ♥❣❤✐Ư♠ ❝đ❛ af (x) + f ′ (x) ❧➭ ♥❤÷♥❣ sè t❤ù❝✳ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✾ ❱Ý ❞ô ✶✳✷✳✶✷✳ ●✐➯ t❤✐Õt t✃t ❝➯ ❝➳❝ ♥❣❤✐Ư♠ ❝đ❛ ➤❛ t❤ø❝ n f (x) ✈➭ ➤❛ t❤ø❝ g(x) = n−1 a0 x + a1 x + · · · + an ➤Ò✉ ❧➭ ♥❤÷♥❣ sè t❤ù❝✳ ❑❤✐ ➤ã t✃t ❝➯ ❝➳❝ ♥❣❤✐➟♠ ′ (n) ❝ñ❛ F (x) = a0 f (x) + a1 f (x) + · · · + an f (x) ❝ị♥❣ ➤Ị✉ ❧➭ ♥❤÷♥❣ sè t❤ù❝✳ g(x) = a0 (x + λ1 )(x + λ2 ) (x + λn ) ✈í✐ ❝➳❝ λj t❤ù❝✳ ′ ′ ❑ý ❤✐Ö✉ F0 (x) = a0 f (x), F1 (x) = F0 (x) + λ1 F0 (x) = a0 [f (x) + λ1 f (x)], F2 (x) = F1 (x)+λ2 F1′ (x) = a0 [f (x)+(λ1 +λ2 )f ′ (x)]+λ1 λ2 f ′′ (x)],✈✳✈✳✳✳ ❝✉è✐ ′ ′ (n) ❝ï♥❣ Fn (x) = Fn−1 (x) + λn Fn−1 (x) = a0 f (x) + a1 f (x) + · · · + an f (x) ❚❤❡♦ ❱Ý ❞ô ✶✳✷✳✶✶ s✉② r❛ t✃t ❝➯ ❝➳❝ ♥❣❤✐Ư♠ ❝đ❛ F0 , F1 , , Fn ➤Ị✉ t❤ù❝✳ ❇➭✐ ❣✐➯✐✿ ❇✐Ĩ✉ ❞✐Ơ♥ f = cos u + C1n cos(u + α)x + · · · + Cnn cos(u + nα)xn ●✐➯✐ ♣❤➢➡♥❣ tr×♥❤ f (x) = ❱Ý ❞ơ ✶✳✷✳✶✸✳ ❈❤♦ ❇➭✐ ❣✐➯✐✿ ➜➷t g = sin u + C1n sin(u + α)x + · · · + Cnn sin(u + nα)xn ❑❤✐ ➤ã f + ig f − ig z t = = = = z + C1n ztx + · · · + Cnn ztn xn = z(1 + tx)n n z + C1n ztx + · · · + Cnn zt xn = z(1 + tx)n cos u + i sin u cos α + i sin α 2f = z(1 + tx)n + z(1 + tx)n f (x) = t➢➡♥❣ + tx z n ➤➢➡♥❣ ✈í✐ z(1 + tx) + z(1 + tx)n = ❤❛② = − = −z + tx z + tx n + tx ◆❤➢ ✈❐② = = cos(2u + π) + i sin(2u + π) ✈➭ ➤➢ỵ❝ + tx + tx 2u + π + k2π 2u + π + k2π ) + i sin( ) ✈í✐ k = 0, 1, , n − ❚õ ➤ã cos( n n ợ x ó P trì n a1 , , an , b ∈ R \ {0} ✈➭ α1 , , αn ❧➭ ♥❤÷♥❣ sè t❤ù❝ n a2k ❝❤Ø ❝ã ♥❣❤✐Ư♠ t❤ù❝✳ ♣❤➞♥ ❜✐Öt✳ ❑❤✐ ➤ã f (x) = b + x − α k k=1 ❱Ý ❞ơ ✶✳✷✳✶✹✳ ●✐➯ sư n a2 (c − α − id) a2k k k ❇➭✐ ❣✐➯✐✿ ❚❛ ❝ã f (c + id) = b + = b+ 2 k=1 c + id − αk k=1 (c − αk ) + d n a2k P❤➬♥ ➯♦ Im(f (c + id)) = −d = ❦❤✐ d = ❱❐② f (c + + b2 (a − α ) k k=1 id) = ❦❤✐ d = ❑❤➠♥❣ ❤➵♥ ❝❤Õ ❝ã t❤Ó ❝♦✐ α1 < α2 < · · · < αn−1 < αn ❍✐Ó♥ ♥❤✐➟♥ f (x) = ❝ã n − ♥❣❤✐Ö♠ t❤ù❝ γk t❤á❛ ♠➲♥ n α1 < γ1 < α2 < γ2 < · · · < αn−1 < γn−1 < αn S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✵ γ t❤á❛ ♠➲♥ ❤♦➷❝ γ ∈ (−∞, α1 ) (αn , +∞) ❚õ ➤ã s✉② r❛ ❤➭♠ f (x) ❝❤Ø ❝ã ❝➳❝ ♥❣❤✐Ư♠ t❤ù❝✳ ✈➭ t❤➟♠ ➤ó♥❣ ♠ét ♥❣❤✐Ư♠ ❱Ý ❞ơ ✶✳✷✳✶✺✳ ❈❤♦ ➤❛ t❤ø❝ γ ∈ P (x) = + x2 + x9 + xn1 + + xns + x1992 ♥❤✐➟♥ ❝❤♦ tr➢í❝ t❤á❛ ♠➲♥ < n1 < < ns < n1 , , ns ❧➭ ❝➳❝ sè tù 1992✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ♥❣❤✐Ư♠ √ ✈í✐ ❤♦➷❝ ❝đ❛ ➤❛ t❤ø❝ P✭①✮ ✭♥Õ✉ ❝ã ✮ ❦❤➠♥❣ t❤Ĩ ❧í♥ 1− ❤➡♥ ✳ ❱Ý ❞ô ✶✳✷✳✶✻✳ ❈❤♦ ➤❛ t❤ø❝ P (x) = x3 − 9x2 + 24x − 97 ✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ỗ số tồ t ột sè ♥❣✉②➟♥ n ❞➢➡♥❣ an s❛♦ ❝❤♦ P (an ) ❝❤✐❛ ❤Õt ❝❤♦ ✳ S hóa b i Trung tâm H c li u – i h c Thái Nguyờn DeThiMau.vn http://www.lrc-tnu.edu.vn ỗ ũ từ ❤×♥❤ t❤ø❝ ◆❤➢ ♠ét sù t✐Õ♣ tơ❝ ❝đ❛ ✈➭♥❤ ➤❛ tứ t ứ ỗ ỹ từ ì t❤ø❝ ♠ét ❜✐Õ♥ tr➟♥ tr➢ê♥❣ ✷✳✶ k = Q, R, C ỗ ũ từ ì tứ ụ t tr ứ ỗ ỹ từ ì t❤ø❝ ♠ét ❜✐Õ♥ tr➟♥ ♠ét tr➢ê♥❣✳ ❑ý ❤✐Ö✉ ∞ k[[x]] = {a0 + a1 x + a2 x + Ã Ã Ã | k} = ỗ tư f ∈ k[[x]], f = t❤õ❛ ❤×♥❤ t❤ø❝ ❝đ❛ ❜✐Õ♥ ∞ x i ✈í✐ x0 = 1, i=0 xi | ∈ k ➤➢ỵ❝ ❣ä✐ ❧➭ ột ỗ ỹ i=0 x ệ tử tộ k ➜Ó ❜✐Õ♥ k[[x]] t❤➭♥❤ ♠ét ✈➭♥❤ ∞ x i , g = ❣✐❛♦ ❤♦➳♥ ❝ã ➤➡♥ ✈Þ t❛ ❝➬♥ ❝➳❝ ♣❤Ð♣ t♦➳♥ s❛✉✳ ❈❤♦ f= k[[x]] t❛ ➤Þ♥❤ ♥❣❤Ü❛ f = g ❝❤♦ ♠ä✐ i=0 i=0 f +g = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ∞ i = bi (ai + bi )x , f g = i=0 ∞ ∞ bi xi ∈ i = 0, 1, ✈➭ i ( ai−j bj )xi i=0 j=0 ▼Ư♥❤ ➤Ị ✷✳✶✳✶✳ ❱í✐ ❝➳❝ ♣❤Ð♣ t♦➳♥ tr➟♥✱ k[[x]] ❧❐♣ t❤➭♥❤ ♠ét ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❝ã ➤➡♥ ✈Þ✳ ❈❤ø♥❣ ♠✐♥❤✿ ❱✐Ư❝ ❦✐Ĩ♠ tr❛ ❝➳❝ t✐➟♥ ➤Ị ❝đ❛ ✈➭♥❤ ❧➭ t❤á❛ ♠➲♥✳ ❚r♦♥❣ ✈➭♥❤ ♥➭② t❛ ❦❤➠♥❣ q✉❛♥ t➞♠ tí✐ tÝ♥❤ ộ tụ tí trị ủ ỗ ỉ q t➞♠ tí✐ tÝ♥❤ ❤÷✉ tØ ✈➭ ❝➠♥❣ t❤ø❝ ➤ã♥❣ ❝đ❛ ỗ t S húa b i Trung tâm H c li u – i h c Thỏi Nguyờn DeThiMau.vn http://www.lrc-tnu.edu.vn tứ ó ủ ỗ ể ♥❣❤✐➟♥ ❝ø✉ tỉ♥❣ ❤❛② ❝➳❝ ❤Ư sè ❝đ❛ ❜✐Ĩ✉ ❞✐Ơ♥ ỗ ì tứ ủ tử f = xi ❧➭ f ′ = iai xi−1 ❱í✐ ♠ét i=0 i=1 ❤➭♠ f (x) ❜✃t ❦ú ①➳❝ ➤Þ♥❤ t➵✐ x = 0, t❛ ❜✐Ĩ✉ ❞✐Ơ♥ ♥ã q ỗ ũ từ ì f (n) (0) tứ f (x) = xn n! n=0 ➜Þ♥❤ ❧ý ✷✳✶✳✷✳ ỗ ũ từ ì tứ f = ỉ a0 = ứ ỗ ỹ từ ì tứ ∞ x i i=0 f (x) = ∞ ❧➭ ➢í❝ ❝đ❛ ➤➡♥ ✈Þ ❦❤✐ xi ❧➭ ➤➡♥ ✈Þ ủ i=0 ỉ tồ t ỗ ỹ t❤õ❛ ❤×♥❤ t❤ø❝ ∞ g(x) = bi xi k[[x]] s❛♦ ❝❤♦ i=0 i f (x)g(x) = ➜✐Ò✉ ♥➭② t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ❤Ư a0 b0 = 1, ai−j bj = ❝❤♦ j=0 ♠ä✐ i = 1, 2, bj ỗ ệ ❣✐➯✐ ➤➢ỵ❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ a0 = g(x) ợ ọ ị ủ f (x) ❦❤✐ ✈✐Õt g(x) = f (x) ◆❣➢ê✐ t❛ tờ q t ế tí ữ tỉ ủ ỗ ệ tử ỗ p(x) q(x) f (x)q(x) = p(x) tr♦♥❣ k[[x]] ◆Õ✉ q(0) = 1, ❜❐❝ ❝ñ❛ f (x) ❧➭ deg f (x) := deg p(x) − deg q(x) ◆Õ✉ tå♥ t➵✐ ❤➭♠ ✭➤➵✐ sè ❤♦➷❝ s✐➟✉ ✈✐Öt✮ F (x) s❛♦ ❝❤♦ f (x) = F (x) t❤× F (x) ợ ọ tứ ó ủ ỗ f (x) trể f (x) ợ ọ ỗ ❤÷✉ tØ ♥Õ✉ ❝ã p(x), q(x) ∈ k[x] ➤Ĩ f (x) = t ỗ ỹ từ ì tứ ột số ❤➭♠ ➤➡♥ ❣✐➯♥ s❛✉ ➤➞②✿ = + x + x2 + x3 + · · · + xn + · · · 1−x xn x x2 x3 x + + ··· + + ··· e = 1+ + 1! 2! 3! n! n x2 x3 x4 n−1 x ln(1 + x) = x − + − + · · · + (−1) + ··· n 2n−1 x3 x5 n−1 x + − · · · + (−1) + ··· sin x = x − 3! 5! (2n − 1)! 2n x2 x4 n x cos x = − + − · · · + (−1) + ··· 2! 4! (2n)! S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✸ arcsin x = arccos x = arctan x = π2 = x3 1.3x5 1.3.5 (2n − 1)x2n+1 x+ + + ··· + + ··· 2.3 2.4.5 2.4.6 (2n).(2n + 1) x3 1.3x5 1.3.5 (2n − 1)x2n+1 π − x+ + + ··· + + ··· 2.3 2.4.5 2.4.6 (2n).(2n + 1) x2n−1 x3 x5 + − · · · + (−1)n−1 + ··· x− 2n − 1 1 + + + ··· + + ··· n ❈❤ó ý ✷✳✶✳✸✳ ➜✐Ị✉ ❦✐Ư♥ ❝❤♦ x ➤Ĩ ❝ã ❜✐Ĩ✉ ❞✐Ơ♥ ♥❤➢ tr➟♥ ❦❤➠♥❣ ①Ðt ë ➤➞②✳ ➜Þ♥❤ ❧ý ✷✳✶✳✹✳ ❬❊✉❧❡r❪ ❱í✐ ♠ä✐ sè t❤ù❝ ❈❤ø♥❣ ♠✐♥❤✿ x t❛ ❝ã eix = cos x + i sin x (ix)n ix (ix)2 (ix)3 + + + ··· + + ··· ❚õ e = + 1! 2! 3! n! ix t❛ s✉② r❛ ➤å♥❣ ♥❤✃t t❤ø❝ x2n x2 x4 + − · · · + (−1)n + ···) 2! 4! (2n)! 2n−1 x3 x5 n−1 x + i(x − + − · · · + (−1) + · · · ) 3! 5! (2n − 1)! eix = (1 − ❉♦ ➤ã eix = cos x + i sin x ❍Ư q✉➯ ✷✳✶✳✺✳ ❱í✐ ♠ä✐ sè t❤ù❝ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ ix iy e e =e i(x+y) eix = ei(x−y) eiy ✈➭ ✈➭ e ix n x, y t❛ ❧✉➠♥ ❝ã ❝➳❝ ❤Ö t❤ø❝ s❛✉ ➤➞②✿ = einx eix = e−ix = ✈í✐ ♠ä✐ n ∈ Z eix eix − e−ix eix + e−ix , sin x = cos x = 2i ❈❤ø♥❣ ♠✐♥❤✿ ❙✉② r❛ tõ ➜Þ♥❤ ❧ý ✷✳✶✳✹✳ ❇ỉ ➤Ị ✷✳✶✳✻✳ ❚❛ ❝ã 1 + + ··· + + ··· 2! 3! n! (−1)n−1 1 + · · · ) π = 4(1 − + − · · · + 2n − e = 2+ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✹ x x2 x3 xn ❈❤ø♥❣ ♠✐♥❤✿ ❱× e = + + + +···+ +··· 1! 2! 3! n! 1 + + ··· + + ··· ❝ã e = + 2! 3! n! x2n−1 x3 x5 + − · · · + (−1)n−1 +··· ❱× arctan x = x − 2n − 1 (−1)n−1 ❝ã π = 4(1 − + − ··· + + · · · ) 2n − x ➜Þ♥❤ ❧ý ✷✳✶✳✼✳ ❙è ♥➟♥ ❦❤✐ ❝❤♦ x=1 ♥➟♥ ❦❤✐ ❝❤♦ x=1 e ❧➭ sè ✈➠ tØ✳ 1 e = + + + ··· + + · · · ●✐➯ sö 2! 3! n! p 1 p + ··· + + + ··· e ❧➭ sè ❤÷✉ tØ✱ e = ❑❤✐ ➤ã = + q q 2! q! (q + 1)! 1 1 ❱❐② p(q − 1)! − (2 + + · · · + )q! = + + ··· < 2! q! q + (q + 1)(q + 2) 1 + ➜✐Ò✉ tể ợ ì ế tr số + · · · = q + (q + 1)2 q ❝ß♥ ✈Õ ♣❤➯✐ ❧➭ ♠ét ♣❤➞♥ sè t❤ù❝ sù ♥❤á ❤➡♥ ✶✳ ❱❐② e ❧➭ sè ✈➠ tØ✳ ứ ỗ ex t ứ sè ❝ã π ❝ò♥❣ ❧➭ sè ✈➠ tû✳ ❑Õt q✉➯ ợ t ể q ị ý s ứ ë ➤➞②❃ ➜Þ♥❤ ❧ý ✷✳✶✳✽✳ ❙è π ❧➭ sè ✈➠ tØ✳ 2n+1 n=0 (2n + 1) sè ♥❣✉②➟♥ ❞➢➡♥❣ m ❱Ý ❞ô ✷✳✶✳✾✳ ❚Ý♥❤ ∞ (2x)2n+2 2n n=0 (2n + 1) n ∞ 2n n 2n n k + ❉♦ ❜ë✐ kak = b0 ak−1 + b1 ak−2 + · · · + bk−2 a1 + bk−1 ✈➭ |kak | > k(k + 1) ♥➟♥ ●✐➯ sö |an | |b0 ak−1 + b1 ak−2 + · · · + bk−2 a1 + bk−1 | |ak−1 | + |ak−2 | + · · · + |a1 | + k(k + 1) < |kak | k + (k − 1) + · · · + + = k(k + 1) : ♠➞✉ t❤✉➱♥✳ ◆❤➢ ✈❐② |an | n + ✈í✐ ♠ä✐ n ✈➭ ♥❤➢ t❤Õ 1 ❞➢➡♥❣ 2 (22 ) 22 · · · (2n ) 2n < ❱Ý ❞ơ ✷✳✶✳✶✷✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ n ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ n ❇➭✐ ❣✐➯✐✿ ❱× ❱× 2 22 n (2 ) · · · (2 ) 2n =2 k=1 k 2k n ♥➟♥ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ∞ ∞ ∞ n k k = = = ♥➟♥ < k r k−1 k k=1 r=k k=1 k=1 k=1 ∞ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn k < k k=1 http://www.lrc-tnu.edu.vn ✶✻ lim ❱Ý ❞ơ ✷✳✶✳✶✸✳ ❚×♠ n→+∞ an , tr♦♥❣ ➤ã ❞➲② sè ♥❣✉②➟♥ (an ) ✈➭ (bn ) t❤á❛ ♠➲♥✿ bn a−1 = 0, b−1 = 1, a0 = 1, b0 = ✈➭ ✈í✐ n : an = 2an−1 + (2n − 1)2 an−2 , bn = 2bn−1 + (2n − 1)2 bn−2 n ❇➭✐ ❣✐➯✐✿ ❇➺♥❣ q✉② ♥➵♣ t❤❡♦ (2k + 1) n t❛ ♥❤❐♥ ➤➢ỵ❝ ❝➳❝ ❝➠♥❣ t❤ø❝ bn = k=0 an an−1 (−1)n (2k + 1) ❱❐② = + ✈í✐ ✈➭ an = (2n + 1)an−1 + (−1) bn bn−1 2n + k=0 1 (−1)n an ♠ä✐ sè ♥❣✉②➟♥ n ◆❤➢ ✈❐② = − + + ··· + ❈❤✉②Ó♥ bn 2n + an π q✉❛ ❣✐í✐ ❤➵♥ t❛ ➤➢ỵ❝ lim = arctan = n→+∞ bn n n−1 ❱Ý ❞ô ✷✳✶✳✶✹✳ ❈❤♦ ❤❛✐ ❞➲② sè ♥❣✉②➟♥ (an ) ✈➭ (bn ) t❤á❛ ♠➲♥✿ a0 = −1, b0 = an = 2n − 1, bn = −n2 , n ❳➞② ❞ù♥❣ ❤❛✐ ❞➲② ❝➳❝ sè ♥❣✉②➟♥ (An ) ✈➭ (Bn ) ♥❤➢ s❛✉✿ A0 = 0, B0 = 1, A1 = 1, B1 = a1 An+1 = an+1 An + bn An−1 , Bn+1 = an+1 Bn + bn Bn−1 , n ✭✐✮ ✭✐✐✮ ❚Ý♥❤ An , Bn t❤❡♦ n An ∈ Q \ Z Bn ❈❤ø♥❣ ♠✐♥❤ Bn n→∞ An lim ✭✐✐✐✮ ❚×♠ ✭✐✈✮ ❈❤ø♥❣ ♠✐♥❤ n S hóa b i Trung tâm H c li u – = k=1 k 1− 12 3− 22 5− i h c Thái Nguyên DeThiMau.vn 32 ✳✳ ✳ − ··· (n − 1)2 2n − − 2n − http://www.lrc-tnu.edu.vn ✶✼ n t❛ ♥❤❐♥ ➤➢ỵ❝ ❝➳❝ ❝➠♥❣ t❤ø❝ Bn = n! An = ( )n! k k=1 n An = ✭✐✐✮ ❚❛ ❝ã ∈ Q \ Z Bn k=1 k Bn 1 Bn ✭✐✐✐✮ ❱× = n = lim + = ♥➟♥ lim n→∞ An x→−1 ln(1 + x) An k=1 k n 1 An = = ✭✐✐✐✮ ❉♦ Bn 12 k=1 k 1− 22 3− 32 5− ··· ✳✳ ✳ − (n − 1)2 2n − − 2n − ❇➭✐ ❣✐➯✐✿ n ✷✳✷ ❇➺♥❣ q✉② ♥➵♣ t❤❡♦ ✈➭ ❉➲② ❤✐Ư✉ ❝đ❛ ♠ét ❞➲② ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✶✳ ❈❤♦ ❞➲② sè {an } = {an }n∈N ❉➲② 0, ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❤✐Ư✉ ❝đ❛ ❞➲② {an } an+1 − an , n {Dan }n∈N ✈í✐ Dan = ❱× ❞➲② ❤✐Ư✉ ❝ị♥❣ ❧➭ ♠ét ❞➲② sè ♥➟♥ t❛ ❝ã t❤Ĩ ❧❐♣ ❞➲② ❤✐Ư✉ ❝đ❛ ♥ã ✈➭ ❦ý ❤✐Ư✉ q✉❛ {D an } ❍✐Ó♥ ♥❤✐➟♥ D2 an = Dan+1 − Dan = an+2 − 2an+1 + an ❚æ♥❣ q✉➳t Dk+1 an = Dk an+1 − Dk an ❱Ý ❞ô ✷✳✷✳✷✳ ✈➭ Dk (Dh an ) = Dk+h an ❱í✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ r, ❞➲② (an), tr♦♥❣ ➤ã an = ❤Ö t❤ø❝ Dan = an+1 − an = n r−1 n r , t❤á❛ ♠➲♥ ❱í✐ ❤❛✐ ❞➲② sè {an} ✈➭ {bn} t❛ ❝ã D(ran +sbn) = rDan +sDbn ✈➭ D (ran + sbn) = rDk an + sDk bn ✈í✐ ♠ä✐ sè r, s ✈➭ sè tù ♥❤✐➟♥ k, n ❇ỉ ➤Ị ✷✳✷✳✸✳ k = rDan +sDbn = r(an+1 −an )+s(bn+1 −bn ) k ♥➟♥ ❝ã ♥❣❛② ❦Õt q✉➯ D(ran + sbn ) = rDan + sDbn ❚æ♥❣ q✉➳t D (ran + sbn ) = rDk an + sDk bn ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ❞Ơ ❞➭♥❣ ❜➺♥❣ q✉✐ ♥➵♣ t❤❡♦ k ❈❤ø♥❣ ♠✐♥❤✿ ❱× D(ran +sbn ) S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✽ {an } ◆Õ✉ Dr+1 an = ✈í✐ ♠ä✐ n t❤× r + sè r k ❤➵♥❣ a0 , Da0 , , D a0 ①➳❝ ➤Þ♥❤ ❤♦➭♥ t♦➭♥ t✃t ❝➯ ❝➳❝ D an ✈í✐ ♠ä✐ k, n j j r+1 ➜➷❝ ❜✐Öt✱ ♥Õ✉ ❞➲② sè {bn } t❤á❛ ♠➲♥ D b0 = D a0 ✈➭ D bn = ✈í✐ ♠ä✐ n 0, j r, t❤× an = bn ✈í✐ ♠ä✐ n ❇ỉ ➤Ị ✷✳✷✳✹✳ ❈❤♦ ❞➲② sè ❈❤ø♥❣ ♠✐♥❤✿ ❍✐Ó♥ ♥❤✐➟♥✳ {an } ◆Õ✉ ❝ã ➤❛ t❤ø❝ p(x) ❜❐❝ r t❤á❛ ♠➲♥ an = t❤× Dr+1 an = ✈í✐ ♠ä✐ n ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ Dr+1 an = ➜Þ♥❤ ❧ý ✷✳✷✳✺✳ ❈❤♦ ❞➲② sè p(n) ✈í✐ ♠ä✐ n ✈í✐ ♠ä✐ n t❤× an = n n n n a0 + Da0 + · · · + Ds a0 + · · · + Dr a0 s r p(x) ❜❐❝ r t❤á❛ ♠➲♥ an = p(n) ✈í✐ ♠ä✐ n ❚❛ ❝❤Ø r❛ D an = ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉✐ ♥➵♣ t❤❡♦ r ❑❤✐ r = ❝ã an = p(n) = a ❱❐② D1 an = a − a = ●✐➯ sư ❦Õt ❧✉❐♥ ➤ó♥❣ ❝❤♦ r − ✈➭ p(x) = cr xr + · · · + c0 ❱× an = p(n) ✈í✐ ♠ä✐ n ♥➟♥ Dan = an+1 − an = p(n + 1) − p(n) ➜➷t q(x) = p(x + 1) − p(x) t❤á❛ ♠➲♥ Dan = q(n) ❱× q(x) r ❧➭ ➤❛ t❤ø❝ ❜❐❝ r − ♥➟♥ D (Dan ) = t❤❡♦ ❣✐➯ t❤✐Õt q✉✐ ♥➵♣✳ ❱❐② t❛ ♥❤❐♥ r+1 ➤➢ỵ❝ D an = r+1 ●✐➯ t❤✐Õt ❞➲② {an } t❤á❛ ♠➲♥ D an = ✈í✐ ♠ä✐ n ➜Þ♥❤ ♥❣❤Ü❛ ❞➲② ♠í✐ {bn } ①➳❝ ➤Þ♥❤ ❜ë✐✿ ❈❤ø♥❣ ♠✐♥❤✿ ●✐➯ sö ➤❛ t❤ø❝ r+1 bn = n n n n a0 , n a0 + · · · + Dr a0 + D a0 + · · · + D s r s ❚❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✸ t❛ ❝ã ♥❣❛② n n n a0 a0 + · · · + Dr+1 a0 + D2 r n n n D r a0 Da0 + D2 a0 + · · · + Dr+1 r−1 Dbn = D = ✈× Dr+1 a0 = ▲➷♣ ❧➵✐✱ ✈í✐ D2 , , Dj D j bn = ✈➭ t❛ ♥❤❐♥ ➤➢ỵ❝ n n n Dj a0 + Dj+1 a0 + · · · + Dr a0 r−j n = Dr a0 ❉♦ ➤ã Dr+1 bn = Dr+1 a0 = ✈í✐ ♠ä✐ Dr bn = Dr a0 r−r n ✈➭ Dj b0 = Dj a0 ✈í✐ ♠ä✐ j r ❱❐② t❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✹ ❝ã an = bn ✈í✐ ♠ä✐ n ✈➭ ➤Õ♥ S hóa b i Trung tâm H c li u – i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn