Đa thức và nội dung dạy học đa thức trong chương trình toán 7, toán 8

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❍❖⑨◆● ❚❍➚ ❑❍⑩◆❍ ▲■◆❍ ✣❆ ❚❍Ù❈ ❱⑨ ◆❐■ ❉❯◆● ❉❸❨ ❍➴❈ ✣❆ ❚❍Ù❈ ❚❘❖◆● ❈❍×❒◆● ❚❘➐◆❍ ❚❖⑩◆ ✼✱ ❚❖⑩◆ ✽ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❍❖⑨◆● ❚❍➚ ❑❍⑩◆❍ ▲■◆❍ ✣❆ ❚❍Ù❈ ❱⑨ ◆❐■ ❉❯◆● ❉❸❨ ❍➴❈ ✣❆ ❚❍Ù❈ ❚❘❖◆● ❈❍×❒◆● ❚❘➐◆❍ ❚❖⑩◆ ✼✱ ❚❖⑩◆ ✽ ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè ❑❍➶❆ ▲❯❾◆ ❚➮❚ P ữớ ữợ ì ❚❍➚ ▲❯❨➌◆ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽ ▲❮■ ❈❷▼ ❒◆ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ✈ỵ✐ ✤➲ t➔✐ ✧✣❛ t❤ù❝ ✈➔ ♥ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽✧✱ ♥❣♦➔✐ sü ❝è ❣➢♥❣ ❝õ❛ ❜↔♥ t❤➙♥✱ ❡♠ ❝á♥ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤ï ❝õ❛ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ ✈➔ ❜↕♥ ❜➧✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❝ỉ ❣✐→♦ ✲ ❚❤❙✳ ❉÷ì♥❣ ❚❤à ▲✉②➳♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï✱ ❝❤➾ ❞➝♥ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ õ ỗ tớ t ỡ ❇❛♥ ❣✐→♠ ❤✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ qỵ t ổ t t ❞↕② ✈➔ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❉♦ ❝❤÷❛ ❝â ❦✐♥❤ ♥❣❤✐➺♠ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ❦❤ỉ♥❣ tr ọ ỏ t sõt ữủ õ ỵ sỷ ỳ rt ữủ ỳ ỵ ❦✐➳♥ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ▲❮■ ❈❆▼ ✣❖❆◆ ❑❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❜↔♥ t❤➙♥ ữợ sỹ ữợ t t ❝õ❛ ❝ỉ ❣✐→♦ ❚❤❙✳ ❉÷ì♥❣ t❤à ▲✉②➳♥✳ ❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✤➣ ❣❤✐ tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❊♠ ①✐♥ ❦❤➥♥❣ ✤à♥❤ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ✏✣❛ t❤ù❝ ✈➔ ♥ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽✑ ❧➔ ❦➳t q✉↔ ❝õ❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ ❤å❝ t➟♣ ✈➔ ♥é ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥✱ ❦❤ỉ♥❣ ❝â sü trò♥❣ ❧➦♣ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ◆➳✉ s❛✐ ❡♠ ①✐♥ ❤♦➔♥ t♦➔♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ✶ ✶ ◆ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽ ✶✳✶ ◆ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽ ✶✳✷ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ❑❤→✐ ♥✐➺♠ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ❇➟❝ ❝õ❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✸ ❈→❝ ♣❤➨♣ t♦→♥ ✈➲ ✤❛ t❤ù❝ ✶✳✶✳✹ ◆❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✷✳✶ ❱➔♥❤ ✤❛ t❤ù❝ ♠ët ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✷✳✷ ❇➟❝ ❝õ❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✳✷✳✸ ◆❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ✸✹ ✶✳✷✳✹ ❳➙② ❞ü♥❣ ✈➔♥❤ ✤❛ t❤ù❝ ♥❤✐➲✉ ➞♥ ✣❛ t❤ù❝ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷ ❍➺ t❤è♥❣ ❤â❛ ❝→❝ ❞↕♥❣ ❜➔✐ t➟♣ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽ ✸✽ ✷✳✶ ❈→❝ ❜➔✐ t♦→♥ ✈➲ t➼♥❤ t♦→♥✱ t➻♠ ❣✐→ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✳✶ ✸✽ ❘ót ❣å♥ ❜✐➸✉ t❤ù❝ ✈➔ t➼♥❤ ❣✐→ trà ❜✐➸✉ t❤ù❝ ✳ ✳ ✳ ✳ ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✷✳✷ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ✷✳✶✳✷ ❚➼♥❤ ♥❤❛♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ tọ tự trữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✶✳✹ P❤÷ì♥❣ ♣❤→♣ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✶✳✺ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t✱ ❧ỵ♥ ♥❤➜t ❝õ❛ ♠ët ❜✐➸✉ t❤ù❝ ✹✶ ❈→❝ ❜➔✐ t♦→♥ ✈➲ ❝❤ù♥❣ ♠✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✷✳✶ x ❈❤ù♥❣ ♠✐♥❤ ❣✐→ trà ❜✐➸✉ t❤ù❝ ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ❣✐→ trà ❝õ❛ ❜✐➳♥ ✷✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✷✳✷ ❈❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✷✳✸ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ❈→❝ ❜➔✐ t♦→♥ ✈➲ ♣❤➨♣ ❝❤✐❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✸✳✶ ❈❤✐❛ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✤➣ s➢♣ ①➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✸✳✷ ❚➻♠ sè ♥❣✉②➯♥ ❜✐➸✉ t❤ù❝ ✷✳✹ B(n) n ✤➸ ❜✐➸✉ t❤ù❝ A(n) ❝❤✐❛ ❤➳t ❝❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ f (x) ❝❤✐❛ ❤➳t ❝❤♦ g(x) ✹✻ ✷✳✸✳✸ ❚➻♠ ❝→❝ ❤➺ sè ✤➸ ✤❛ t❤ù❝ ✳ ✹✼ ✷✳✸✳✹ ❚➻♠ ❞÷ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✳✺ ⑩♣ ❞ö♥❣ ✈➔♦ sè ❤å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✸✳✻ ▼ët sè ❤➡♥❣ ✤➥♥❣ t❤ù❝ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ❈→❝ ❜➔✐ t♦→♥ ✈➲ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû ✳ ✳ ✳ ✳ ✺✶ ✷✳✹✳✶ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✹✳✷ ⑩♣ ❞ö♥❣ ✈➔♦ sè ❤å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ số trữợ (x, y) t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✹✳✹ P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷✳✹✳✺ P❤÷ì♥❣ ♣❤→♣ ❤➺ sè ❜➜t ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✐✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✷✳✹✳✻ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ỵ t t tự r❛ t❤ø❛ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✼ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➨t ❣✐→ trà r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❳➙② ❞ü♥❣ ❤➺ t❤è♥❣ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠ ✸✳✶ ✸✳✷ ✺✹ ỳ ữ ỵ ỹ ❤➺ t❤è♥❣ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✶✳✶ ❱➲ ♥ë✐ ❞✉♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✶✳✷ ❱➲ ❤➻♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ❍➺ t❤è♥❣ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠ ✈➲ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ❑➳t ❧✉➟♥ ✼✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✼✸ ✐✐✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ▲❮■ ▼Ð ✣❺❯ ✣❛ t❤ù❝ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ q✉❛♥ trå♥❣ ❝õ❛ t♦→♥ ❤å❝✱ ổ s ữợ t ỏ tt ữợ tr t ợ ữủ t sợ r ữỡ tr tổ t ♥ë✐ ❞✉♥❣ ✈➲ ✤❛ t❤ù❝ ✤÷đ❝ ❣✐↔♥❣ ❞↕② t➟♣ tr✉♥❣ ð ❜➟❝ tr✉♥❣ ❤å❝ ❝ì sð✳ ❑❤→✐ ♥✐➺♠ ✤❛ t❤ù❝ ❜➢t ✤➛✉ ✤÷đ❝ ✤÷❛ ✈➔♦ ❝❤÷ì♥❣ ■❱ ♣❤➛♥ ✣↕✐ sè ❧ỵ♣ ✼✳ ❈→❝ ✈➜♥ ✤➲ ❝ì ❜↔♥ ❦❤→❝ ✈➲ ✤❛ tự ữủ t ố ữỡ tr ợ ỳ ❜➔✐ t♦→♥ ✈➲ ✤❛ t❤ù❝ t❤÷í♥❣ ①✉➜t ❤✐➺♥ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ q✉è❝ ❣✐❛✱ ❖❧②♠♣✐❝ q✉è❝ t➳ ✈➔ ❧✉ỉ♥ ✤÷đ❝ ✤→♥❤ ❣✐→ ❧➔ ❜➔✐ t♦→♥ ❦❤â✳ ❍✐➺♥ ♥❛② ♠ỉ♥ ❚♦→♥ ✤➣ ✤÷đ❝ ❝❤✉②➸♥ s❛♥❣ ❤➻♥❤ t❤ù❝ t❤✐ tr➢❝ ♥❣❤✐➺♠ ❝❤♦ ❦ý t❤✐ ①➨t tèt ♥❣❤✐➺♣ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ✈➔ t✉②➸♥ s✐♥❤ ✤↕✐ ❤å❝✱ ❝❛♦ ✤➥♥❣✳ ✣➸ ✤↕t ✤÷đ❝ ❦➳t q✉↔ ❝❛♦ t❤➻ ❤å❝ s✐♥❤ ❝➛♥ ✤÷đ❝ ❧➔♠ q✉❡♥ ✈ỵ✐ ❤➻♥❤ t❤ù❝ ♥➔② ❝➔♥❣ sỵ♠ ❝➔♥❣ tèt✳ ✣➦❝ ❜✐➺t ❧➔ ♥❣÷í✐ ❞↕② ❝➛♥ ♣❤↔✐ ❝â sü ❧ü❛ ❝❤å♥ ♣❤÷ì♥❣ ♣❤→♣✱ ❝→❝❤ t❤ù❝ ❞↕② ❤å❝ ♣❤ò ❤đ♣ ✈➔ ❤➺ t❤è♥❣ ❜➔✐ t➟♣ ✤❛ ❞↕♥❣✱ ♣❤♦♥❣ ♣❤ó ✤è✐ ✈ỵ✐ ♠å✐ ✤è✐ t÷đ♥❣ ❤å❝ s✐♥❤✳ ❉♦ ✤â ✤➸ ❣✐↔♥❣ ❞↕② ✤÷đ❝ tèt ♥ë✐ ❞✉♥❣ ✈➲ ✤❛ t❤ù❝ ❝❤♦ ❤å❝ s✐♥❤ t❤➻ ♥❣÷í✐ ❣✐→♦ ✈✐➯♥ ❝➛♥ ♣❤↔✐ ❝â ❦✐➳♥ t❤ù❝ ✈➲ ✤❛ t❤ù❝ ♠ët ❝→❝❤ ✤➛② ✤õ ❝ơ♥❣ ♥❤÷ ❝➛♥ ❤✐➸✉ s➙✉ ✈➲ ♥ë✐ ❞✉♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❞↕② ❤å❝ ✤❛ t❤ù❝ tổ ỗ tớ õ ✈➲ ♠ö❝ ✤➼❝❤ ❞↕② ❤å❝ ❝õ❛ ♣❤➛♥ ✤❛ t❤ù❝✳ ✣➸ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ✤✐➲✉ tr➯♥ t❤➻ ♠ët ❦ÿ ♥➠♥❣ ❦❤ỉ♥❣ t t ố ợ ữớ t ữỡ tr s ợ ỳ ỵ tr ụ ợ ỏ s ự ữủ sỹ ❣✐ó♣ ✤ï✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ ❚❤❙✳ ❉÷ì♥❣ ❚❤à ▲✉②➳♥ ❡♠ ✤➣ ♠↕♥❤ ❞↕♥ ❝❤å♥ ✤➲ t➔✐✿ ✧✣❛ t❤ù❝ ✈➔ ♥ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ t♦→♥ ✼✱ t♦→♥ ✽✧ ✤➸ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥❤➡♠ ♣❤➙♥ t➼❝❤ ♥ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ ❝ò♥❣ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ♥â tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ✣↕✐ sè ❧ỵ♣ ✼✱ ❧ỵ♣ ✽✳ ❚ø ✤â ❝â t❤➸ sû ❞ö♥❣ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝õ❛ t♦→♥ ❝❛♦ ❝➜♣ ✤➸ s♦✐ s→♥❣ ❧↕✐ ✈➔ ♣❤➙♥ ❧♦↕✐✱ ❤➺ t❤è♥❣ ♠ët sè ❜➔✐ t♦→♥ ✈➲ ✤❛ t❤ù❝ ❝ơ♥❣ ♥❤÷ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â tr♦♥❣ ♠ỉ♥ t♦→♥ ð ♥❤➔ tr÷í♥❣ ♣❤ê t❤ỉ♥❣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ◆ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽✳ ❈❤÷ì♥❣ ✷✳ ❍➺ t❤è♥❣ ❤â❛ ❝→❝ ❞↕♥❣ ❜➔✐ t➟♣ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽✳ ❈❤÷ì♥❣ ✸✳ ❳➙② ❞ü♥❣ ❤➺ t❤è♥❣ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠✳ ❉♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ♥➠♥❣ ❧ü❝ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ s❛✐ sât✳ ❊♠ r➜t ♠♦♥❣ ✤÷đ❝ sỹ õ ỵ t ổ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✷ ❈❤÷ì♥❣ ✶ ◆ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽ ✶✳✶ ◆ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽ ◆ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✤❛ t❤ù❝ ❜➢t ✤➛✉ ✤÷đ❝ tr➻♥❤ ❜➔② ❝❤➼♥❤ t❤ù❝ ữỡ s số ợ t ❝❤✐➳♠ t❤í✐ ❧÷đ♥❣ ❧➔ ✷✵ t✐➳t ✈➔ ❝❤✐➳♠ 25% ❧÷đ♥❣ ❦✐➳♥ t❤ù❝✳ ❈→❝ ✈➜♥ ✤➲ ❝ì ❜↔♥ ❝õ❛ ♥â ✤÷đ❝ t ố tr ữỡ s số ợ t➟♣ ✶✱ ❝❤✐➳♠ t❤í✐ ❧÷đ♥❣ ❧➔ ✷✶ t✐➳t ✈➔ ❝❤✐➳♠ 30% ❧÷đ♥❣ ❦✐➳♥ t❤ù❝✳ ❉♦ ✤â ✈✐➺❝ ❞↕② ♥ë✐ ❞✉♥❣ ❦✐➳♥ t❤ù❝ ✈➲ ✤❛ t❤ù❝ t❤❡♦ ♠ët ❝→❝❤ ♥➔♦ ✤â ✤➸ ✤↕t ❤✐➺✉ q✉↔ ❝❛♦ ♥❤➜t ❝õ❛ ♥❣÷í✐ ❤å❝ ❧➔ ✈➜♥ ✤➲ ❝➛♥ t❤✐➳t ❝õ❛ ♥❣÷í✐ ❣✐→♦ ✈✐➯♥✳ ✣➸ ❝â ♣❤÷ì♥❣ t❤ù❝ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ t❤❡♦ ♠ët ❝→❝❤ ❤✐➺✉ q✉↔ t❤➻ ♥ë✐ ❞✉♥❣ t❛ ❝➛♥ ♣❤➙♥ t➼❝❤ ✤➛✉ t✐➯♥ ❧➔ ❦❤→✐ ♥✐➺♠ ✤❛ t❤ù❝✳ ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼ ❝â ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ✤❛ t❤ù❝ ✈➔ ❦❤→✐ ♥✐➺♠ ✤❛ t❤ù❝ ♠ët ❜✐➳♥✱ ❦❤→✐ ♥✐➺♠ ✤❛ t❤ù❝ ✤÷đ❝ tr trữợ tự ởt ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ữợ từ t➟♣ tr➢❝ ♥❣❤✐➺♠ ❦❤→ rë♥❣✳ ❙û ❞ö♥❣ tr➢❝ ♥❣❤✐➺♠ ✤↔♠ ❜↔♦ t➼♥❤ ❦❤→❝❤ q✉❛♥ ❦❤✐ ❝❤➜♠ ✤✐➸♠✱ ❣➙② ✤÷đ❝ t➼♥❤ ❤ù♥❣ t❤ó ✈➔ t➼♥❤ t➼❝❤ ❝ü❝ ❤å❝ t➟♣ ❝õ❛ ❤å❝ s✐♥❤✱ ❤å❝ s✐♥❤ ❝â t❤➸ tü ✤→♥❤ ❣✐→ ❜➔✐ ❧➔♠ ❝õ❛ ♠➻♥❤ ✈➔ t❤❛♠ ❣✐❛ ✤→♥❤ ❣✐→ ❜➔✐ ❧➔♠ ❝õ❛ ỳ ữ ỵ ỹ ❤➺ t❤è♥❣ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠ ✸✳✶✳✶ ❱➲ ♥ë✐ ❞✉♥❣ ❈→❝ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠ ❝➛♥ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ②➯✉ ❝➛✉ ❝ì ❜↔♥ s❛✉ ✤➙②✿ ✲ ❇❛♦ q✉→t ✤÷đ❝ ♠ët ❝→❝❤ t♦➔♥ ❞✐➺♥ ❝→❝ ♥ë✐ ❞✉♥❣ ❝õ❛ ❜➔✐✱ ❝õ❛ ❝❤÷ì♥❣✳ ✲ ✣→♥❤ ❣✐→ t♦➔♥ ❜ë ❝→❝ ♠ö❝ t✐➯✉ ✈➲ ❦✐➳♥ t❤ù❝ ✈➔ ❦ÿ ♥➠♥❣ ✤➣ q✉② ✤à♥❤ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤✳ ✲ ❈❤➾ r❛ ✤÷đ❝ ❝→❝ s❛✐ ❧➛♠ t❤÷í♥❣ ♠➢❝ ♣❤↔✐ ❝õ❛ ❤å❝ s✐♥❤✳ ✸✳✶✳✷ ❱➲ ❤➻♥❤ t❤ù❝ ❈→❝ ❜➔✐ t➟♣ ❝➛♥ ✤÷đ❝ ✤❛ ❞↕♥❣ ❤â❛ ✈➲ ❞↕♥❣ ❜➔✐✱ tr→♥❤ tr÷í♥❣ ❤đ♣ r❛ q✉→ ♥❤✐➲✉ ❜➔✐ ð ❝ò♥❣ ♠ët ❞↕♥❣ tr♦♥❣ ❝ò♥❣ ♠ư❝ t✐➯✉ ❤å❝ ❣➙② ♥❤➔♠ ❝❤→♥✱ ♠➜t ❤ù♥❣ t❤ó ✤è✐ ✈ỵ✐ ❤å❝ s✐♥❤✳ ✸✳✷ ❍➺ t❤è♥❣ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠ ✈➲ ✤❛ t❤ù❝ ❈➙✉ ✶✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ t➼♥❤ (x − 1)2 ❆✳ x2 − 2x + 1✳ ❇✳ x2 + 2x + 1✳ ✻✵ t❛ ✤÷đ❝✿ ❈✳ (x−1)(x+1)✳ ❉✳ x2 + 2x − 1✳ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❈➙✉ ✷✳ ❱✐➳t ❜✐➸✉ t❤ù❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ x3 + 9x2 + 27x + 27 ữợ ❧➟♣ ♣❤÷ì♥❣ ❝õ❛ ♠ët tê♥❣ t❛ ❝â✿ ❆✳ (x − 3)3 ✳ ❇✳ (x + 3)3 ✳ ❈✳ x3 − 33 ✳ ❉✳ ❈➙✉ ✸✳ ❑❤✐ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐❛ (2x5 − 8x4 + 6x2 y ) : 2x2 ❦➳t q✉↔ ❧➔✿ ❆✳ x3 − 4x2 + 3y ✳ ❇✳ x3 − 4x2 + 3xy ✳ ❈✳ x3 − 4x2 + 3y ✳ ❉✳ x3 + 4x2 + 3y ✳ ❈➙✉ ✹✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ t➼♥❤ 252 − 152 ❆✳ 10✳ ❇✳ 40✳ x3 + 3 ✳ ❧➔✿ ❈✳ 100✳ ❉✳ 400✳ ❈➙✉ ✺✳ ●✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ 5x(4x2 − 2x + 1) − 2x(10x2 − 5x − 2) t↕✐ x = 15 ❆✳ ❧➔✿ 135✳ ❇✳ 125✳ ❈✳ 115✳ ❉✳ 155✳ ❉✳ x2 + 6x + 9✳ ❈➙✉ ✻✳ P❤➨♣ ♥❤➙♥ (x − 3)(x2 + 3x + 9) ❝â ❦➳t q✉↔ ❧➔✿ ❆✳ x3 + 27✳ ❇✳ x3 − 27✳ ❈✳ x2 − 6x + 9✳ ❈➙✉ ✼✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ t♦→♥ (31, 8)2 − 2.31, 8.21, + (21, 8)2 ❆✳ 102✳ ❇✳ 100✳ ❈✳ 200✳ ❧➔✿ ❉✳ 10✳ ❉✳ 1✳ ❉✳ 2✳ ❈➙✉ ✽✳ ●✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ x2 − 20x + 101 ❧➔✿ ❆✳ 20✳ ❇✳ 101✳ ❈✳ 11✳ ❈➙✉ ✾✳ ●✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ 4x − x2 + ❧➔✿ ❆✳ 4✳ ❈➙✉ ✶✵✳ ❇✳ 3✳ ❈❤♦ sè tü ♥❤✐➯♥ n ❈✳ 7✳ ❝❤✐❛ ❝❤♦ ✼ ❞÷ ✹✳ ❍ä✐ n2 ❝❤✐❛ ❝❤♦ ✼ ❞÷ ❜❛♦ ♥❤✐➯✉❄ ❆✳ 0✳ ❇✳ 1✳ ❈✳ 2✳ ❉✳ 3✳ ❉✳ x = 2✳ ❈➙✉ ✶✶✳ ❇✐➳t 2x(x − 5) − x(2x + 3) = 26✱ ❣✐→ trà ❝õ❛ x ❧➔✿ ❆✳ x = −2✳ ❈➙✉ ✶✷✳ ❇✳ x = 5✳ ❈✳ x = −5✳ ❈â ❤❛✐ ❤➻♥❤ ❝❤ú ♥❤➟t✳ ❍➻♥❤ t❤ù ♥❤➜t ❝â ❝❤✐➲✉ ❞➔✐ ❤ì♥ ❝❤✐➲✉ rë♥❣ ✾♠✳ ❍➻♥❤ t❤ù ❤❛✐ ❝â ❝❤✐➲✉ rë♥❣ ❤ì♥ ❝❤✐➲✉ rë♥❣ ❤➻♥❤ t❤ù ♥❤➜t ❧➔ ✻✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ✺♠ ✈➔ ❝â ❝❤✐➲✉ ❞➔✐ ❤ì♥ ❝❤✐➲✉ ❞➔✐ ❤➻♥❤ t❤ù ♥❤➜t ❧➔ ✶✺♠✳ ❇✐➳t ❞✐➺♥ t➼❝❤ ❤➻♥❤ t❤ù ❤❛✐ ❤ì♥ ❞✐➺♥ t➼❝❤ ❤➻♥❤ t❤ù ♥❤➜t ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t t❤ù ♥❤➜t✱ ❣✐→ trà ❝õ❛ ❆✳ x = 9✳ ❇✳ x 640m2 ✳ ●å✐ x ❧➔ ❝❤✐➲✉ rë♥❣ ❧➔✿ x = 25✳ ❈✳ x = 26✳ ❉✳ x = 15✳ ❈➙✉ ✶✸✳ ❈❤♦ a + b = 1✳ ❚➼♥❤ ❣✐→ trà M = 2(a3 + b3 ) − 3(a2 + b2 )✳ ❆✳ M = 1✳ ❇✳ M = −1✳ ❈✳ M = −2✳ ❉✳ M = 2✳ ❈➙✉ ✶✹✳ ❳→❝ ✤à♥❤ a✱ b✱ c ❜✐➳t✿ (ax2 + bx + c)(x + 3) = x3 + 2x2 − 3x ✈ỵ✐ ♠å✐ x✳ ❆✳ a = 1✱ b = −1✱ c = 0✳ ❇✳ a = 1✱ b = 2✱ x = 0✳ ❈✳ a = 1✱ b = −2✱ c = 1✳ ❉✳ a = 1✱ b = −1✱ c = 1✳ ❈➙✉ ✶✺✳ ❈❤♦ ✤❛ t❤ù❝✿ f (x) = x(x + 1)(x + 2)(ax + b)✳ ❳→❝ ✤à♥❤ a✱ b ✤➸ f (x) − f (x − 1) = x(x + 1)(2x + 1) ✈ỵ✐ ❆✳ a = ✱ b = 1✳ ❈✳ a = 1✱ b = ✳ ❈➙✉ ✶✻✳ ❚➻♠ x ❜✐➳t✿ (x − 3)2 − = 0✳ ♠å✐ x✳ 1 a=− ✱b= ✳ 2 1 ❉✳ a = ✱ b = ✳ 2 ❇✳ ❆✳ x = 5✱ x = 1✳ ❇✳ x = 4✱ x = 3✳ ❈✳ x = −5✱ x = 1✳ ❉✳ x = −4✱ x = 3✳ ❈➙✉ ✶✼✳ ●✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ x − x2 ❆✳ ✳ ❈➙✉ ✶✽✳ ❧➔✿ ✳ ❈✳ 2✳ x(x − y)2 + y(y − 3x)2 ❝ò♥❣ ❇✳ ❇✐➸✉ t❤ù❝ ❉✳ 4✳ ✈ỵ✐ ❜✐➸✉ t❤ù❝ ♥➔♦ s❛✉ ✤➙② t↕♦ ♥➯♥ ✤➥♥❣ t❤ù❝❄ ❆✳ (x + y)3 ✳ ❈➙✉ ✶✾✳ ❚➻♠ y ❆✳ y= ✳ ❇✳ (x − y)3 ✳ ❈✳ ❉✳ x3 − y ✳ (3y − y + 1)(y − 1) + y (4 − 3y) = ✳ ❇✳ y = ✳ ❈✳ y = ✳ ❉✳ y = ✳ 4 2n 2n−1 1−2n 2−2n ♣❤➨♣ t➼♥❤ (2x + 3x )(x − 3x ) t❛ ✤÷đ❝✿ ❜✐➳t✿ ❈➙✉ ✷✵✳ ❚❤ü❝ ❤✐➺♥ x3 + y ✳ ✻✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ❆✳ −6x2 − 7x + 3✳ ❇✳ 6x2 + 7x + ❈✳ −6x2 + 7x + 3✳ ❉✳ 6x2 − 7x + 3✳ ❈➙✉ ✷✶✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ s❛✉ t❤➔♥❤ ♥❤➙♥ tû✿ 2x3 + 3x2 y + 2xy ✳ ❆✳ x(2x2 + 3xy + 2y)✳ ❈✳ 2x(x + 3xy + 2y)✳ ❈➙✉ ✷✷✳ xy(2x2 + 3x + 2) ❉✳ 2xy x + x + ✳ P t↕✐ x = 2020✱ y = 2018✳ P = ❇✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ x(x − 2017) + y(2017 − x)✳ ❆✳ 16✳ ❇✳ 8✳ ❈✳ 18✳ 6✳ ❉✳ ❈➙✉ ✷✸✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ s❛✉ t❤➔♥❤ ♥❤➙♥ tû✿ 15x3 − 5x2 + 10x✳ ❆✳ 3x(5x2 − x + 2)✳ ❇✳ 5x(3x2 − x + 2) ❈✳ x(3x2 − x + 2)✳ ❉✳ 2x(5x2 − x + 5)✳ ❈➙✉ ✷✹✳ ❚➻♠ x ❜✐➳t✿ (x − 1)(x2 + x) + 2(x − 1) = 0✳ ❆✳ x = 1✳ ❇✳ x = −1✳ ❈✳ x = ±1✳ x = 2✳ ❉✳ ❈➙✉ ✷✺✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ t➼♥❤ 37, 5.6, − 7, 5.3, − 6, 6.7, + 3, 5.37, ❧➔✿ ❆✳ 300✳ ❇✳ 350✳ ❈➙✉ ✷✻✳ ●✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ ❆✳ 3✳ ❈➙✉ ✷✼✳ ❇✳ 370✳ 432 − 112 (36, 5)2 − (27, 5)2 ❈✳ 4✳ ❚➻♠ ❝→❝ ❝➦♣ sè ♥❣✉②➯♥ ❈✳ (x, y) ❉✳ 375✳ ❉✳ 6✳ ❧➔✿ 5✳ t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝ xy + 3x − 2y − = ❆✳ (1.4) ❈✳ (1, −4) ✈➔ (3 − 2)✳ ❇✳ (1, −4) ✈➔ (3, −2)✳ (3, 2)✳ ❉✳ (1, −4) ✈➔ (−3, 2)✳ ✈➔ ❈➙✉ ✷✽✳ ❚➻♠ x ❜✐➳t✿ (2x − 1)2 − 25 = 0✳ ❆✳ x = −3✱ x = −2✳ ❇✳ x = −3✱ x = 2✳ ❈✳ x = 3✱ x = −2✳ ❉✳ x = 3✱ x = 2✳ ❈➙✉ ✷✾✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ s❛✉ t❤➔♥❤ ♥❤➙♥ tû✿ (1 + x2 )2 − 4x(1 − x2 )✳ ✻✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ❆✳ (1 − x2 − 2x)2 ✳ ❇✳ (1 − x2 + 2x)2 ✳ ❈✳ (1 + x2 − 2x)2 ✳ ❉✳ (1 + x2 + 2x)2 ✳ ❈➙✉ ✸✵✳ ●✐→ trà ❝õ❛ ❆✳ 16299✳ ❈➙✉ ✸✶✳ ●✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ y = 1, 4✱ z = 1, ❆✳ 973 + 833 − 97.83 ❧➔✿ ❜✐➸✉ t❤ù❝ 180 ❇✳ 16290✳ ❈✳ 16295✳ ❇✳ ❈❤♦ A = x(2x − y) − (y − 2x) 16298✳ ✈ỵ✐ x = 1, 2✱ ❧➔✿ 2✳ ❈➙✉ ✸✷✳ ❉✳ 5✳ a + b + c = 0✳ ❈✳ ❇✐➸✉ t❤ù❝ 3✳ a3 + b3 + c3 ❉✳ 7✳ ❝ò♥❣ ✈ỵ✐ ❜✐➸✉ t❤ù❝ ♥➔♦ s❛✉ ✤➙② t↕♦ ♥➯♥ ✤➥♥❣ t❤ù❝❄ ❆✳ 3abc✳ ❇✳ 3a2 bc✳ ❈✳ 3ab2 c✳ ❉✳ 3abc2 ✳ ❈➙✉ ✸✸✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ x2 − x − t❤➔♥❤ ♥❤➙♥ tû✳ ❆✳ (x + 3)(x + 2)✳ ❇✳ (x + 3)(x − 2)✳ ❈✳ (x − 3)(x + 2)✳ ❉✳ (x − 3)(x − 2)✳ ❈➙✉ ✸✹✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ x2 − 2xy + y − xz + yz t❤➔♥❤ ♥❤➙♥ tû✳ ❆✳ (x + y)(x − y − z)✳ ❇✳ (x − y)(x + y + z)✳ ❈✳ (x + y + z)(x + y)✳ ❉✳ (x − y − z)(x − y)✳ ❈➙✉ ✸✺✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ ❝❤✐❛ 258 : 512 ❧➔✿ ❆✳ 25✳ ❇✳ 125✳ ❈✳ 625✳ ❉✳ 5✳ ❈➙✉ ✸✻✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ ❝❤✐❛ (5x4 − 2x3 + x2 ) : 2x2 ❧➔✿ x +x+ ✳ 2 x −x− ✳ 2 ❇✳ x −x+ ✳ ❆✳ 2 ❈✳ ❉✳ − x + x + ✳ 2 4 ❈➙✉ ✸✼✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ (−15x y z ) : (5x y z ) t↕✐ x = − ✱ y = − ❧➔✿ ❆✳ −2✳ ❇✳ 2✳ ❈✳ −3✳ ❉✳ 3✳ ❈➙✉ ✸✽✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ ❝❤✐❛ (x5 + x3 + x2 + 1) : (x3 + 1) ❧➔✿ ✻✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❆✳ x2 + ✳ ❇✳ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ x + 1✳ ❈✳ (x + 1)2 ✳ ❉✳ x − 1✳ ❉✳ x + 2✳ ❈➙✉ ✸✾✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ ❝❤✐❛ (x2 + 5x + 6) : (x + 3) ❧➔✿ ❆✳ x − 2✳ ❇✳ x2 + ✳ ❈✳ x2 − 2✳ ❈➙✉ ✹✵✳ ❳→❝ ✤à♥❤ a ✤➸ ✤❛ t❤ù❝ 2x3 − 3x2 + x + a ❝❤✐❛ ❤➳t ❝❤♦ ✤❛ t❤ù❝ x + 2✳ ❆✳ a = 10✳ ❈➙✉ ✹✶✳ ❝❤♦ ❇✳ a = 20✳ ❈✳ a = 30✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ n ❉✳ 2n2 + 3n + ✤➸ 2n − 1✳ ❆✳ n ∈ {0; 1; −2; 3}✳ ❇✳ n ∈ {0; 1; 2; 3}✳ n ∈ {0; 1; 2; −3}✳ ❉✳ n ∈ {0; −1; 2; 3}✳ ❈✳ a = 40✳ ❝❤✐❛ ❤➳t ❈➙✉ ✹✷✳ ❉÷ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ ✤❛ t❤ù❝ f (x) = 2x5 − 703 + 4x2 − x + ❝❤♦ x−6 ❧➔✿ ❆✳ 570✳ ❇✳ 571✳ ❈✳ 572✳ ❉✳ 573✳ ❈➙✉ ✹✸✳ ❈❤♦ ✤❛ t❤ù❝ f (x)✱ ❝→❝ ♣❤➛♥ ❞÷ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ f (x) ❝❤♦ x ✈➔ x − ❧➛♥ ❧÷đt ❧➔ ✶ ✈➔ ✷✳ ❚➻♠ ♣❤➛♥ ❞÷ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ f (x) ❝❤♦ x(x − 1)✳ ❆✳ x − 1✳ ❈➙✉ ✹✹✳ ❇✳ ❇✐➳t ✤❛ t❤ù❝ x − 1✳ ❈✳ x + 1✳ f (x) = 3x3 − 7x2 + 4x − ❚❤÷ì♥❣ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ f (x) ❝❤♦ x−2 ❉✳ x2 + 1✳ ❝❤✐❛ ❤➳t ❝❤♦ x − 2✳ ❧➔✿ ❆✳ 3x2 + x + 2✳ ❇✳ 3x2 − x + 2✳ ❈✳ 3x2 + x − 2✳ ❉✳ 3x2 − x − 2✳ ❈➙✉ ✹✺✳ ❳→❝ ✤à♥❤ ❝→❝ ❤➺ sè a ✈➔ b ✤➸ ✤❛ t❤ù❝ f (x) = x4 + ax2 + b ❝❤✐❛ ❤➳t ❝❤♦ g(x) = x2 − 3x + 2✳ ❆✳ a = −5✱ b = 4✳ ❇✳ a = 5✱ b = 4✳ ❈✳ a = 5✱ b = −4✳ ❉✳ a = −5✱ b = −4✳ ❈➙✉ ✹✻✳ ●✐↔ sû ✤❛ t❤ù❝ f (x) = x4 +ax2 +b ❝❤✐❛ ❤➳t ❝❤♦ g(x) = x3 −3x+2✳ ❚❤÷ì♥❣ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ f (x) = x4 + ax2 + b ✻✺ ❝❤♦ g(x) = x3 − 3x + ❧➔✿ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ❆✳ x2 − 3x − 2✳ ❇✳ x2 − 3x + 2✳ ❈✳ x2 + 3x + 2✳ ❉✳ −x2 + 3x + 2✳ ❈➙✉ ✹✼✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ 3x3 − 7x2 + 4x − t❤➔♥❤ ♥❤➙♥ tû✱ t❛ ✤÷đ❝✿ ❆✳ (x − 2)(3x2 + x + 2)✳ ❇✳ (x − 2)(3x2 − x + 2)✳ ❈✳ (x + 2)(3x2 + x + 2)✳ ❉✳ (x + 2)(3x2 − x + 2)✳ ❈➙✉ ✹✽✳ ❉÷ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ ✤❛ t❤ù❝ f (x) = + x2 + x4 + x6 + · · · + x100 ❝❤♦ x+1 ❆✳ ❧➔✿ 51✳ ❇✳ 52✳ ❈✳ 53✳ ❉✳ 54✳ ❈➙✉ ✹✾✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ f (x) = (x2 + x)2 + 3(x2 + x) + t❤➔♥❤ ♥❤➙♥ tû✱ ✤➦t y = x2 + x✱ t❛ ✤÷đ❝✿ ❆✳ (y − 1)(y − 2)✳ ❇✳ (y + 1)(y − 2)✳ ❈✳ (y − 1)(y + 2)✳ ❉✳ (y + 1)(y + 2)✳ ❈➙✉ ✺✵✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ f (x) = x(x + 1)(x + 2)(x + 3) + ♥❤➙♥ tû✱ t❛ ✤÷đ❝✿ ❆✳ (x2 + 3x − 1)2 ✳ ❇✳ (x2 + 3x + 1)2 ✳ ❈✳ (x2 − 3x − 1)2 ✳ ❉✳ (x2 − 3x + 1)2 ✳ ✣→♣ →♥ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ❆ ❇ ❆ ❉ ❆ ❇ ❇ ❉ ❈ ❈ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ❆ ❈ ❇ ❆ ❉ ❆ ❇ ❆ ❉ ❆ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ❆ ❉ ❇ ❆ ❆ ❆ ❇ ❈ ❆ ❉ ✸✶ ✸✷ ✸✸ ✸✹ ✸✺ ✸✻ ✸✼ ✸✽ ✸✾ ✹✵ ❈ ❆ ❈ ❉ ❈ ❇ ❈ ❆ ❉ ❈ ✻✻ t❤➔♥❤ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ✹✶ ✹✷ ✹✸ ✹✹ ✹✺ ✹✻ ✹✼ ✹✽ ✹✾ ✺✵ ❆ ❇ ❈ ❇ ❆ ❈ ữợ ởt số ❤ä✐ tr➢❝ ♥❣❤✐➺♠ ❈➙✉ ✹✳ 252 − 152 = (25 − 15).(25 + 15) = 10.40 = 400✳ ❈➙✉ ✺✳ 5x(4x2 − 2x + 1) − 2x(10x2 − 5x − 2) = 20x3 − 10x2 + 5x − 20x3 + 10x2 + 4x = 9x ❚❤❛② x = 15 t❛ ✤÷đ❝ ❦➳t q✉↔ ❧➔ 9.15 = 135✳ ❈➙✉ ✻✳ (x − 3)(x2 + 3x + 9) = (x − 3)(x2 + 3x + 32 ) = (x − 3)(x2 + 3x + 32 ) = x3 − 33 = x3 − 27 ❈➙✉ ✼✳ (31, − 21, 8)2 = 100✳ ❈➙✉ ✽✳ x2 − 20x + 101 = (x − 10)2 + ≥ 1✳ ❈➙✉ ✾✳ −(x2 − 4x + 4) + = − (x − 2)2 ≤ 7✳ ❈➙✉ ✶✵✳ n = 7x + ⇒ n2 = (7x + 4)2 = 7(7x2 + 8x + 2) + 2✳ ❈➙✉ ✶✷✳ ❚❤❡♦ ✤➲ ❜➔✐✱ t❛ ❝â✿ (x+5)(x+9+15)−x(x+9) = 640 ⇒ x = 26✳ ❈➙✉ ✶✻✳ (x−3)2 −4 = (x−3)2 −22 = (x−3−2)(x−3+2) = (x−5)(x−1)✳ 1 ❈➙✉ ✶✼✳ x − x = −(x − x + ) = − x − 4 2 ✻✼ ≤ ✳ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ❈➙✉ ✶✽✳ x(x − 3y)2 + y(y − 3x)2 = x(x2 − 6xy + 9y ) + y(y − 6xy + 9x2 ) = x3 − 6x2 y + 9xy + y − 6xy + 9x2 y = x3 + 3x2 y + 3xy + y = (x + y)3 ❈➙✉ ✷✶✳ 2x3 + 3x2 y + 2xy = x(2x2 + 3xy + 2y)✳ ❈➙✉ ✷✷✳ P = x(x − 2017) + y(2017 − x) = (x − 2017)(x − y)✳ ❚❤❛② sè✱ P = (2020 − 2017)(2020 − 2018) = 3.2 = 6✳ ❈➙✉ ✷✹✳ (x − 1)(x2 + x) + 2(x − 1) = (x − 1)(x2 + x + 2)✳ ❱➻ x + x + = x+ + > ∀x✳ ❑❤✐ ✤â (x − 1)(x + x + 2) = ❦❤✐ x − = 0✳ ❙✉② r❛ x = 1✳ ❈➙✉ ✷✺✳ 37, 5.6, − 7, 5.3, − 6, 6.7, + 3, 5.37, =(37, 5.6, + 3, 5.37, 5) − (7, 5.3, + 6, 6.7, 5) =37, 5(6, + 3, 5) − 7, 5.(3, + 6, 6) =37, 5.10 − 7, 5.10 = 375 − 75 = 300 432 − 112 (43 − 11)(43 + 11) 32.54 ❈➙✉ ✷✻✳ = = = 3✳ (36, 5)2 − (27, 5)2 (36, − 27, 5)(36, + 27, 5) 9.64 ❈➙✉ ✷✼✳ ✰✮ xy + 3x − 2y − = xy + 3x − 2y − − = (x − 2)(y + 3) − 1✳ ✰✮ (x − 2)(y + 3) − = ⇔ (x − 2)(y + 3) = ✻✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝     x − = −1    y + = −1  ⇔   x − =    y + = ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤     x=1    y = −4  ⇔   x =    y = −2 ❈➙✉ ✷✽✳ (2x − 1)2 − 25 = (2x − 1)2 − 52 = (2x − − 5)(2x − + 5) = (2x − 6)(2x + 4)✳ ❈➙✉ ✷✾✳ ⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ (a + b)2 = (a − b)2 + 4ab✳ ❚❛ ❝â✿ (1 + x2 )2 − 4x(1 − x2 ) = (1 − x2 )2 + 4x2 − 4x(1 − x2 ) = (1 − x2 − 2x)2 ❈➙✉ ✸✵✳ 973 + 833 (97 + 83)(972 − 97.83 + 832 ) − 97.83 = − 97.83 180 180 = 972 + 832 = 16298 ❈➙✉ ✸✶✳ A = x(2x − y) − z(y − 2x) = (2x − y)(x + z)✳ ❱ỵ✐ x = 1, 2✱ y = 1, 4✱ z = 1, t❤➻ A = 3✳ ❈➙✉ ✸✷✳ a + b + c = ⇒ c = −(a + b) ⇒ c3 = −(a + b)3 ❚❤❛② c3 = −(a + b)3 ✈➔♦ a3 + b3 + c3 ✳ ❈➙✉ ✸✸✳ x2 −x−6 = x2 −3x+2x−6 = x(x−3)+2(x−3) = (x−3)(x+2)✳ ❈➙✉ ✸✹✳ x2 −2xy +y −xz +yz = (x−y)2 −(x−y)z = (x−y)(x−y −z)✳ ❈➙✉ ✸✽✳ x5 + x3 + x2 + = x3 (x2 + 1) + (x2 + 1) = (x2 + 1)(x3 + 1)✳ ❈➙✉ ✸✾✳ x2 + 5x + = x2 + 2x + 3x + = (x + 2)(x + 3)✳ ✻✾ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ❈➙✉ ✹✵✳ f (x) ❝❤✐❛ ❤➳t ❝❤♦ x + ❦❤✐ f (−2) = 0✱ tù❝ 2(−2)3 − 3(−2)2 + (−2) + a = ⇒ a = 30 ❈➙✉ ✹✶✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐❛ 2n +3n+3 ❝❤♦ 2n−1 t❛ ✤÷đ❝ ✳ 2n − 2n2 + 3x + 2n − 2n2 + 3n + = 2n − n+2+ ✣➸ 2n − ❧➔ sè ♥❣✉②➯♥ t❤➻ 2n − ♣❤↔✐ số r ữợ ìợ ỗ số ợ 2n = −1✱ ❱ỵ✐ 2n − = 1✱ ❱ỵ✐ 2n − = −5✱ ❱ỵ✐ 2n − = 5✱ ❱➟② ✈ỵ✐ ❈➙✉ ✹✷✳ t❛ ❝â t❛ ❝â n = 0❀ n = 1❀ t❛ ❝â t❛ ❝â ±1✱ ±5✳ n = −2❀ n = 3✳ n = 0; 1; −2; t❤➻ 2n2 + 3n + ❝❤✐❛ ❤➳t ỷ sỡ ỗ rr ữ tr 2n − 1✳ f (x) ❝❤♦ x−6 ❧➔ f (6) = 571✳ ✻ ❈➙✉ ✹✸✳ ✷ ✵ −70 ✹ −1 ỵ ❇❡③♦✉t t❛ ❝â ❝â ❜➟❝ ❤❛✐ ♥➯♥ ❞÷ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ ✶✳ ●✐↔ sû ❞÷ ❧➔ r(x) = ax + b f (0) = f (x) ❝❤♦ ✈➔ f (1) = 2✳ x(x − 1) ❚❤❛② x=0 ✈➔♦ ✭✶✮ t❛ ✤÷đ❝ f (0) = b = 1✳ ❚❤❛② x=1 ✈➔♦ ✭✶✮ t❛ ✤÷đ❝ f (1) = a + b = 2✳ a = b = 1✳ x(x − 1) ❝â ❜➟❝ ❦❤æ♥❣ q✉→ t❛ ❝â✿ f (x) = x(x − 1)q(x) + ax + b ❚ø ✤â s✉② r❛ ❱➻ ❱➟② ❞÷ ❝➛♥ t➻♠ ❧➔ ✼✵ x + 1✳ ✭✶✮ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ❈➙✉ ✹✹✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐❛✿ 3x3 −7x2 +4x −4 3x3 −6x2 x−2 3x2 − x + −x2 +4x −4 −x2 +2x 2x −4 2x −4 ✵ ❈➙✉ ✹✺✳ g(x) = x2 − x + 2x + = x(x − 1) − 2(x − 1) = (x − 1)(x − 2)✳ ◆➳✉ ❝❤♦ f (x) x − 2✳ ❚❤❛② ❝❤✐❛ ❤➳t ❝❤♦ g(x) t❤➻ f (x) ❚❤❡♦ ỵ t t õ t f (1) = ✈➔ x−1 ✈➔ ❝❤✐❛ ❤➳t f (2) = 0✳ x = 1✱ x = ✈➔♦ f (x) t❛ ✤÷đ❝✿ + a + b = ✈➔ 16 + 4a + b = 0✳ ❚ø ✤â s✉② r❛ a = −5✱ b = 4✳ ❈➙✉ ✹✻✳ ⑩♣ ❞ư♥❣ ❦➳t q✉↔ ❝➙✉ ✹✺✱ ①→❝ ✤à♥❤ ✤÷đ❝ a = −5✱ b = 4✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐❛ ✤❛ t❤ù❝ ✤÷đ❝ t❤÷ì♥❣ f (x) = x4 −5x2 +4 ❝❤♦ ✤❛ t❤ù❝ x2 −3x+2 x2 + 3x + 2✳ ❈➙✉ ✹✼✳ ⑩♣ ❞ö♥❣ ❦➳t q✉↔ ❝➙✉ ✹✹✱ ❦❤✐ ✤â✿ 3x2 − 7x2 + 4x − = (x − 2)(3x2 − x + 2) ỷ sỡ ỗ rr ❞÷ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ f (−1) = 51✳ ✼✶ f (x) ❝❤♦ x+2 ❧➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ ❈➙✉ ✹✾✳ ✣➦t y = x2 + x t❛ ❝â✿ (x2 + x)2 + 3(x2 + x) + = y + 3y + = (y + y)(2y + 2) = y(y + 1) + 2(y + 1) = (y + 1)(y + 2) ❈➙✉ ✺✵✳ ❚❛ ❝â✿ x(x + 1)(x + 2)(x + 3) + = [x(x + 3)] [(x + 1)(x + 2)] + = (x2 + 3x)(x2 + 3x + 2) + ✣➦t x2 + 3x = y ✱ t❛ ❝â✿ (x2 + 3x)(x2 + 3x + 2) + = y(y + 2) + = y + 2y + = (y + 1)2 = (x2 + 3x + 1)2 ✼✷ ❑➳t ❧✉➟♥ ✣❛ t❤ù❝ ❝â ✈à tr➼ q✉❛♥ trå♥❣ tr♦♥❣ ❚♦→♥ ❤å❝✱ ❦❤ỉ♥❣ ♥❤ú♥❣ ❧➔ ✤è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝❤õ ②➳✉ ❝õ❛ ✤↕✐ sè ♠➔ ❝á♥ ❧➔ ❝ỉ♥❣ ❝ư ✤➢❝ ❧ü❝ ❝õ❛ ❣✐↔✐ t➼❝❤✳ ◆â ❧➔ ♣❤➛♥ ❦✐➳♥ t❤ù❝ q✉❛♥ trå♥❣ ✤÷đ❝ ❣✐ỵ✐ t❤✐➺✉ ♥❣❛② tø ♥❤ú♥❣ ♥➠♠ ✤➛✉ ❝õ❛ ❜➟❝ ♣❤ê tổ r ỵ tt tự ỏ ữủ sỷ ❞ö♥❣ ♥❤✐➲✉ tr♦♥❣ t♦→♥ ❝❛♦ ❝➜♣✱ t♦→♥ ù♥❣ ❞ö♥❣✳ ❱➔ ❝❤ó♥❣ t❛ t❤÷í♥❣ ①✉②➯♥ ❣➦♣ ♥❤ú♥❣ ❜➔✐ t♦→♥ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✱ t❤✐ ❖❧②♠♣✐❝ t♦→♥ ❤å❝ tr♦♥❣ tr÷í♥❣ ♣❤ê t❤ỉ♥❣✳ ❚✉② ❦❤â❛ ❧✉➟♥ ♥➔② tr➻♥❤ ❜➔② ❦✐➳♥ t❤ù❝ ✈➲ ✤❛ t❤ù❝ ✈➔ ❤➺ t❤è♥❣ ❤â❛ ♥❤ú♥❣ ❜➔✐ t♦→♥ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ tr ỡ s ữ ỏ rt ọ s ợ ❧÷đ♥❣ ❦✐➳♥ t❤ù❝ ✈➲ ✤❛ t❤ù❝✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ♣❤➙♥ t➼❝❤ ♥ë✐ ❞✉♥❣ ❞↕② ❤å❝ ✈➲ ✤❛ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ✼✱ t♦→♥ ✽✱ ❤➺ t❤è♥❣ ❤â❛ ✈➔ ♣❤➙♥ ❧♦↕✐ ❜➔✐ t➟♣ tr♦♥❣ s→❝❤ ❣✐→♦ ❦❤♦❛✱ s❛✉ ✤â ✤÷❛ r❛ ❤➺ t❤è♥❣ ❜➔✐ t➟♣ tr➢❝ ♥❣❤✐➺♠ ❣✐ó♣ ❝❤♦ ♥❣÷í✐ ❞↕② ❝â t❤➸ ✤÷❛ r❛ ✤÷đ❝ sü ❧ü❛ ❝❤å♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❝ơ♥❣ ♥❤÷ ❝→❝❤ t❤ù❝ ❞↕② ❤å❝ ♣❤ò ❤đ♣ ✤➸ ✤↕t ✤÷đ❝ ❤✐➺✉ q✉↔ ❝❛♦ ♥❤➜t✳ ❉♦ ❧➛♥ ✤➛✉ t✐➯♥ ❧➔♠ q✉❡♥ ✈ỵ✐ ❝ỉ♥❣ t→❝ ♥❣❤✐➯♥ ❝ù✉✱ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ r➜t ữủ sỹ õ õ ỵ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✼✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ P❤❛♥ ✣ù❝ ❈❤➼♥❤ ✭✷✵✶✶✮✱ ❙→❝❤ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✼✱ t➟♣ ✷✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✷❪ P❤❛♥ ✣ù❝ ❈❤➼♥❤ ✭✷✵✶✶✮✱ ❙→❝❤ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✽✱ t➟♣ ✶✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✸❪ ❍♦➔♥❣ ❳✉➙♥ ❙➼♥❤ ✭✶✾✾✹✮✱ ❬✹❪ ❚æ♥ ❚❤➙♥ ✭✷✵✶✼✮✱ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣✱ ◆❳❇ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✳ ❈→❝ ❞↕♥❣ t♦→♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❚♦→♥ ✽✱ t➟♣ ✶✱ ◆❳❇ ●✐→♦ ❞ư❝ ❱✐➺t ◆❛♠✳ ❬✺❪ ❚ỉ♥ ❚❤➙♥✱ ✭✷✵✶✶✮✱ ❇➔✐ t➟♣ ❚♦→♥ ✼✱ t➟♣ ✷✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✻❪ ❚æ♥ ❚❤➙♥✱ ✭✷✵✶✶✮✱ ❇➔✐ t➟♣ ❚♦→♥ ✽✱ t➟♣ ✶✱ ◆❳❇ ●✐→♦ ❞ư❝ ❱✐➺t ◆❛♠✳ ❬✼❪ ❇ò✐ ❱➠♥ ❚✉②➯♥ ✭✷✵✶✸✮✱ ❇➔✐ t➟♣ ♥➙♥❣ ❝❛♦ ✈➔ ♠ët sè ❝❤✉②➯♥ ✤➲ ❚♦→♥ ✼✱ ◆❳❇ ●✐→♦ ❞ư❝ ❱✐➺t ◆❛♠✳ ❬✽❪ ❇ò✐ ❱➠♥ ❚✉②➯♥ ✭✷✵✶✸✮✱ ❇➔✐ t➟♣ ♥➙♥❣ ❝❛♦ ✈➔ ♠ët sè ❝❤✉②➯♥ ✤➲ ❚♦→♥ ✽✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ✼✹ ... ❧➔♠ ♥❤÷ s❛✉✿ (2x2 y).(9xy ) = (2.9)(x2 y)(xy ) = 18( x2 x)(yy ) = 18x3 y ✶✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚❛ ♥â✐ ✤ì♥ t❤ù❝ ❍♦➔♥❣ ❚❤à ❑❤→♥❤ ▲✐♥❤ 18x3 y ❧➔ t➼❝❤ ❝õ❛ ❤❛✐ ✤ì♥ t❤ù❝ 2x2 y ✈➔ 9xy ✳ ✯ ◆❤➙♥... +15x2 +11x −3 x2 − 4x − 2x4 −8x3 −6x2 −5x3 +21x2 +11x −3 −5x3 +20x2 +15x x2 2x2 − 5x −4x −3 ❚❤ü❝ ❤✐➺♥ t÷ì♥❣ tü ♥❤÷ tr➯♥✱ t❛ ✤÷đ❝✿ 2x4 −13x3 +15x2 +11x −3 x2 − 4x − 2x4 −8x3 −6x2 2x2 − 5x + −5x3 +21x2... ♥❣♦➦❝ ✭❦➸ ❝↔ ❞➜✉ ❝õ❛ ❝❤ó♥❣✮✳ ❱➼ ❞ư✿ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ s❛✉ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤➦t ♥❤➙♥ tû ❝❤✉♥❣✳ 28x2 y − 21xy + 14x2 y = 7xy(4xy − 3y + 2x) ✰✮ ❉ò♥❣ ❤➡♥❣ ✤➥♥❣ t❤ù❝ ❱➟♥ ❞ư♥❣ ❝→❝ ❤➡♥❣ ✤➥♥❣ t❤ù❝ ✤→♥❣