Bất đẳng thức và cực trị sinh bởi các đa thức đại số ba biến

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

❉×❒◆● ❈➷◆● ❈Ø

❇❻❚ ✣➃◆● ❚❍Ù❈ ❱⑨ ❈Ü❈ ❚❘➚ ❙■◆❍
❇Ð■ ❈⑩❈ ✣❆ ❚❍Ù❈ ✣❸■ ❙➮ ❇❆ ❇■➌◆
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✾


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

❉×❒◆● ❈➷◆● ❈Ø

❇❻❚ ✣➃◆● ❚❍Ù❈ ❱⑨ ❈Ü❈ ❚❘➚ ❙■◆❍
❇Ð■ ❈⑩❈ ✣❆ ❚❍Ù❈ ✣❸■ ❙➮ ❇❆ ❇■➌◆
❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P
▼➣ sè✿

ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝✿

●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ❱➠♥ ▼➟✉

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✾


✐

▼ư❝ ❧ư❝
▼Ð ✣❺❯
❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥

✶
✸

✶✳✶

▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ê ✤✐➸♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝

✶✳✷

✣❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ♠ët sè ❤➺ t❤ù❝ ❝ì ❜↔♥

✶✳✸

✳ ✳ ✳ ✳

✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✷✳✶

❈ỉ♥❣ t❤ù❝ ❱✐➧t❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ✸

✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✷✳✷

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜❛ ➞♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✷✳✸

P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✶✳✷✳✹

❚➼♥❤ ❝❤✐❛ ❤➳t ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣

✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✣❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛

❜✐➳♥
✷✷
✷✳✶

✷✳✷

✷✳✸

❇➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ✤❛ t❤ù❝ ❜➟❝ ❜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✷✳✶✳✶

❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỵ ỡ ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✳ ✳ ✳ ✳

✷✹

❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥

✳ ✳ ✳

✷✽

✷✳✷✳✶

▼ët sè ♠➺♥❤ ✤➲ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✷✳✷✳✷

⑩♣ ❞ö♥❣ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝

✸✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

▼ët sè ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❜❛ ❜✐➳♥ tr♦♥❣ ♣❤➙♥ t❤ù❝

✳ ✳ ✳ ✳

✸✺

❈❤÷ì♥❣ ✸✳ ❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè
❜❛ ❜✐➳♥
✸✽
✸✳✶

❈ü❝ trà t❤❡♦ r➔♥❣ ❜✉ë❝ tê♥❣ ✈➔ t➼❝❤ ❜❛ sè

✸✳✷

❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥

✳

✹✶

✸✳✸

▼ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

❑➌❚ ▲❯❾◆
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✹✼
✹✽


✶

▼ð ✤➛✉
❈❤✉②➯♥ ✤➲ ❜➜t ✤➥♥❣ t❤ù❝ ❝â ✈❛✐ trá r➜t q✉❛♥ trå♥❣ ð ❜➟❝ tr✉♥❣ ❤å❝ ♣❤ê
t❤æ♥❣✳ ❇➜t ✤➥♥❣ t❤ù❝ ❦❤ỉ♥❣ ❝❤➾ ❧➔ ✤è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ trå♥❣ t➙♠ ❝õ❛
✣↕✐ sè ✈➔ ●✐↔✐ t➼❝❤ ♠➔ ❝á♥ ❧➔ ❝ỉ♥❣ ❝ư ✤➢❝ ❧ü❝ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝
❝õ❛ t♦→♥ ❤å❝✳ ❚❛ ✤➣ ❜✐➳t r➡♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✤❛ t❤ù❝ ✤➣ ✤÷đ❝
♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤↔♦ s→t ♥❤÷ ◆❡✇t♦♥✱ ▲❛❣r❛♥❣❡✱ ❇❡rst❡✐♥✱ ▼❛r❦♦✈✱

❑♦❧♠♦❣♦r♦✈✱ ▲❛♥❞❛✉✱ ✳ ✳ ✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❞↕♥❣ ♥➔② ❝ơ♥❣ ❝â t❤➸ ❝❤ù♥❣
♠✐♥❤ ✤÷đ❝ ❜➡♥❣ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❤➻♥❤ ❤å❝ ♥❤÷ ♣❤÷ì♥❣
♣❤→♣ ✈➨❝tì ✈➔ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✱ ♣❤÷ì♥❣ ♣❤→♣ sè ♣❤ù❝✱✳ ✳ ✳
❚✉② ♥❤✐➯♥✱ ❝→❝ ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ù♥❣ ✈ỵ✐ ❧ỵ♣ ✤❛ t❤ù❝ tê♥❣ q✉→t t❤➻ ♥❣÷í✐
t❛ ❝➛♥ ✤➳♥ ❝→❝ ❝ỉ♥❣ ❝ư ❝õ❛ t t ỗ ó st ú
ự ỗ ữù ỗ ữù ❤å❝ s✐♥❤ ❣✐ä✐ ✈➔
♥➙♥❣ ❝❛♦ ♥❣❤✐➺♣ ✈ö ❝õ❛ ❜↔♥ t❤➙♥ ✈➲ ❝❤✉②➯♥ ✤➲ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà
s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✱ tæ✐ ❝❤å♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✧❇➜t ✤➥♥❣
t❤ù❝ ✈➔ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✧✳
▲✉➟♥ ✈➠♥ ♥➔② ♥❤➡♠ ❝✉♥❣ ❝➜♣ ♠ët sè ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà s✐♥❤
❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❝ò♥❣ ♠ët sè q
ỗ t ✈➔

3

❝❤÷ì♥❣✳

❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥✳
❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳
❈❤÷ì♥❣ ✸✳ ❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳
▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ ❦❤↔♦ s→t ♠ët sè ❧ỵ♣ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔
❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✈➔ ①➨t ❝→❝ ♠ð rë♥❣ ❝õ❛ ❝❤ó♥❣
✤➸ →♣ ❞ư♥❣ tr♦♥❣ ❦❤↔♦ s→t ❝→❝ ❜➔✐ t♦→♥ ❝ü❝ trà ❧✐➯♥ q✉❛♥✳
❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ❱➠♥ ▼➟✉
✤➣ t➟♥ t➻♥❤ ữợ ú ù t tr sốt q tr➻♥❤ ❤å❝ t➟♣
✈➔ ♥❣❤✐➯♥ ❝ù✉ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤
tỵ✐ ❝→❝ ❚❤➛② ❈ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥✲❚✐♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝
❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝
t➟♣ t↕✐ ❚r÷í♥❣✳

ỗ tớ t ụ ỷ ớ ỡ tợ ỗ
ổ ổ ú ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ tr♦♥❣ q✉→
tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳

❚❤→✐ ◆❣✉②➯♥✱ ✶✷ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✾✳

❚→❝ ❣✐↔

❉÷ì♥❣ ❈ỉ♥❣ ❈ø


✸

❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝
❧✐➯♥ q✉❛♥
▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ê ✤✐➸♥
❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝ ♥â✐ ❝❤✉♥❣✱ ✤❛ t❤ù❝ ❜➟❝ ❜❛ ♥â✐ r✐➯♥❣ ✈➔ ①➨t ♠ët sè
❤➺ t❤ù❝ ❝ì ❜↔♥✳ ▼ët ♣❤➛♥ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ❞➔♥❤ ✤➸ ♥➯✉ ✈➲ ✤❛ t❤ù❝
❜➟❝ ❜❛ ✈➔ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝
t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✸❪✳

✶✳✶ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ê ✤✐➸♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ A
❈❤♦

❜➟❝

n

❜✐➳♥

x

❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ❚❛ ❣å✐ ✤❛ t❤ù❝

❧➔ ♠ët ❜✐➸✉ t❤ù❝ ❝â ❞↕♥❣

fn (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 (an = 0),
tr♦♥❣ ✤â ❝→❝

ai ∈ A

✤÷đ❝ ❣å✐ ❧➔ ❤➺ sè✱

an

❧➔ ❤➺ sè ❝❛♦ ♥❤➜t ✈➔

a0

✭✶✳✶✮
❧➔ ❤➺ sè

tü ❞♦ ❝õ❛ ✤❛ t❤ù❝✳

fn (x) ❧➔ sè ♠ô ❝❛♦ t ừ ụ tứ õ t tr
ữủ ỵ ❤✐➺✉ ❧➔ deg(f )✳ ❑❤✐ ✤â ♥➳✉ tr♦♥❣ ✭✶✳✶✮ an = 0 t❤➻ deg(f ) = n.
◆➳✉ ai = 0, i = 1, . . . , n ✈➔ a0 = 0 t❤➻ t❛ ❝â ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ ❧➔ 0✳

◆➳✉ ai = 0, i = 0, . . . , n t❤➻ t❛ ❝♦✐ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ ❧➔ −∞ ✈➔ ❣å✐ ✤❛
❇➟❝ ❝õ❛ ✤❛ t❤ù❝

t❤ù❝ ❦❤æ♥❣ ✭♥â✐ ❝❤✉♥❣ t❤➻ ♥❣÷í✐ t❛ ❦❤ỉ♥❣ ✤à♥❤ ♥❣❤➽❛ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝
❦❤ỉ♥❣✮✳ ❚➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ✈ỵ✐ ❤➺ sè tr


A[x]
A=K

A

ữủ ỵ

K[x] ởt ❝â ✤ì♥
✈à✳ ❚❛ t❤÷í♥❣ ①➨t A = Z✱ ❤♦➦❝ A = Q ❤♦➦❝ A = R ❤♦➦❝ A = C✳ ❑❤✐ ✤â✱
t❛ ❝â ❝→❝ ✈➔♥❤ ✤❛ t❤ù❝ t÷ì♥❣ ù♥❣ ❧➔ Z[x], Q[x], R[x], C[x]✳
❑❤✐

❧➔ ♠ët tr÷í♥❣ t❤➻ ✈➔♥❤


✹

❈→❝ ♣❤➨♣ t➼♥❤ tr➯♥ ✤❛ t❤ù❝
❈❤♦ ❤❛✐ ✤❛ t❤ù❝

f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ,
g(x) = bn xn + bn−1 xn−1 + · · · + b1 x + b0 .
❚❛ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ♣❤➨♣ t➼♥❤ sè ❤å❝

f (x) + g(x) = (an + bn )xn + · · · + (a1 + b1 )x + a0 + b0 ,
f (x) − g(x) = (an − bn )xn + · · · + (a1 − b1 )x + a0 − b0 ,
f (x)g(x) = c2n x2n + c2n−1 x2n−1 + · · · + c1 x + c0 ,
tr♦♥❣ ✤â

ck = a0 bk + a1 bk−1 + · · · + ak b0 ,

k = 0, . . . , n.

t t ỡ

ỵ sỷ A ❧➔ ♠ët tr÷í♥❣✱ f (x) ✈➔ g(x) = 0 ❧➔ ❤❛✐ ✤❛ t❤ù❝
A[x]✱ t❤➳
A[x] s❛♦ ❝❤♦

❝õ❛ ✈➔♥❤
t❤✉ë❝

t❤➻ ❜❛♦ ❣✐í ❝ô♥❣ ❝â ❝➦♣ ✤❛ t❤ù❝ ❞✉② ♥❤➜t

f (x) = g(x)q(x) + r(x)
◆➳✉

r(x) = 0

t❛ ♥â✐

f (x)

✈ỵ✐

❝❤✐❛ ❤➳t ❝❤♦

a

❧➔ ♣❤➛♥ tû tũ ỵ ừ

n
ỵ ừ

A[x]

tỷ

f (a) =



r(x)

deg r(x) < deg g(x).

g(x)✳
n

●✐↔ sû

q(x)

A✱ f (x) =

ai x i

❧➔ ✤❛ t❤ù❝ tị②

i=0

ai ai

❝â ✤÷đ❝ ❜➡♥❣ ❝→❝❤ t❤❛②

x

❜ð✐

a

i=0

f (x) t↕✐ a✳
◆➳✉ f (a) = 0 t❤➻ t❛ ❣å✐ a ❧➔ ♥❣❤✐➺♠ ❝õ❛ f (x)✳ ❇➔✐ t♦→♥ t➻♠ ❝→❝ ♥❣❤✐➺♠
❝õ❛ f (x) tr♦♥❣ A ❣å✐ ❧➔ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ❜➟❝ n tr♦♥❣ A✳

✤÷đ❝ ❣å✐ ❧➔ ❣✐→ trà ❝õ❛

an xn + an−1 xn−1 + · · · + a1 x + a0 = 0 (an = 0).

ỵ sỷ A ❧➔ ♠ët tr÷í♥❣✱ a ∈ A ✈➔ f (x) A[x] ữ số ừ

f (x)



xa



f (a)

ỵ ✶✳✸✳ a ❧➔ ♥❣❤✐➺♠ ❝õ❛ f (x) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (x) ❝❤✐❛ ❤➳t ❝❤♦ (x−a)✳
a ∈ A✱ f (x) ∈ A[x] ✈➔ m ❧➔ ♠ët sè tü ♥❤✐➯♥
❤ì♥ ❤♦➦❝ ❜➡♥❣ 1✳ ❑❤✐ ✤â a ❧➔ ♥❣❤✐➺♠ ❜ë✐ ❝➜♣ m ❝õ❛ f (x) ❦❤✐ ✈➔ ❝❤➾
f (x) ❝❤✐❛ ❤➳t ❝❤♦ (x − a)m ✈➔ f (x) ❦❤æ♥❣ ❝❤✐❛ ❤➳t (x a)m+1

sỷ

A

ởt trữớ

ợ



✺
❚r♦♥❣ tr÷í♥❣ ❤đ♣

m = 1 t❤➻ t❛ ❣å✐ a ❧➔ ♥❣❤✐➺♠ ✤ì♥ ❝á♥ ❦❤✐ m = 2 t❤➻ a

✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❦➨♣✳ ❙è ♥❣❤✐➺♠ ❝õ❛ ♠ët ✤❛ t❤ù❝ ❧➔ tê♥❣ sè ❝→❝ ♥❣❤✐➺♠
❝õ❛ ✤❛ t❤ù❝ ✤â ❦➸ ❝↔ ❜ë✐ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ✭♥➳✉ ❝â✮✳ ❱➻ ✈➟②✱ ♥❣÷í✐ t❛ ❝♦✐ ♠ët
✤❛ t❤ù❝ ❝â ♠ët ♥❣❤✐➺♠ ❜ë✐ ❝➜♣

m

♥❤÷ ♠ët ✤❛ t❤ù❝ õ

m

trũ



ữủ ỗ rr
sỷ

f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 A[x]
ợ

A

ởt trữớ õ tữỡ ú ❝õ❛

♠ët ✤❛ t❤ù❝ ❝â ❜➟❝ ❜➡♥❣

n − 1✱

f (x)

❝❤♦

(x − a)

❧➔

❝â ❞↕♥❣

q(x) = bn−1 xn−1 + · · · + b1 x + b0 ,
tr♦♥❣ ✤â

bn−1 = an , bk = abk+1 + ak+1 , k = 0, . . . , n − 2,
✈➔ ❞÷ sè

r = ab0 + a0 .

ỵ



t

sỷ ữỡ tr

an xn + an−1 xn−1 + · · · + a1 x + a0 = 0 (an = 0)
❝â

n

♥❣❤✐➺♠ ✭t❤ü❝ ❤♦➦❝ ♣❤ù❝✮

x1 , x2 , . . . , xn

t❤➻




E1 (x) := x1 + x2 + · · · + xn






E2 (x) := x1 x2 + x1 x3 + · · · + xn−1 xn


.........






En (x) := x1 x2 . . . xn
❜✳ ◆❣÷đ❝ ❧↕✐ ♥➳✉ ❝→❝ sè

x1 , x2 , . . . , xn

✭✶✳✷✮

an−1
=−
an
an−2
=
an
......
a0
= (−1)n .
an

✭✶✳✸✮

t❤ä❛ ♠➣♥ ❤➺ tr➯♥ t❤➻ ❝❤ó♥❣ ❧➔

♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮✳ ❍➺ ✭✶✳✸✮ ❝â
k
t❤➔♥❤ ♣❤➛♥ t❤ù k ❝â Cn sè ❤↕♥❣✳

n

t❤➔♥❤ ♣❤➛♥ ✈➔ ð ✈➳ tr→✐ ❝õ❛

E1 (x), E2 (x), . . . , En (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✭✤❛ t❤ù❝✮ ✤è✐ ①ù♥❣
❜➟❝ 1, 2, . . . , n✱ t÷ì♥❣ ù♥❣✳

❝✳ ❈→❝

sỡ t

ỵ ộ tự tỹ n ✤➲✉ ❝â ❦❤æ♥❣ q✉→ n ♥❣❤✐➺♠ t❤ü❝✳


✻

❍➺ q✉↔ ✶✳✶✳ ✣❛ t❤ù❝ ❝â ✈æ sè ♥❣❤✐➺♠ ❧➔ ✤❛ t❤ù❝ ❦❤æ♥❣✳
❍➺ q✉↔ ✶✳✷✳ ◆➳✉ ✤❛ t❤ù❝ ❝â ❜➟❝ ≤ n ♠➔ ♥❤➟♥ ❝ị♥❣ ♠ët ❣✐→ trà ♥❤÷ ♥❤❛✉
t↕✐

n+1

✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ ✤è✐ sè t❤➻ ✤â ❧➔ ✤❛ t❤ù❝ ❤➡♥❣✳

❍➺ q✉↔ ✶✳✸✳ ❍❛✐ ✤❛ t❤ù❝ ❜➟❝ ≤ n ♠➔ ♥❤➟♥ n + 1 trò♥❣ ♥❤❛✉ t↕✐ n + 1
✤✐➸♠ t ừ ố số t ú ỗ t

ỵ ồ tự f (x) R[x] ❝â ❜➟❝ n ✈➔ ❝â ❤➺ sè ❝❤➼♥❤ ✭❤➺ sè
an = 0

❝❛♦ ♥❤➜t✮

✤➲✉ ❝â t❤➸ ♣❤➙♥ t➼❝❤ ✭❞✉② ♥❤➜t✮ t❤➔♥❤ ♥❤➙♥ tû ❞↕♥❣

m

s

i=1

✈ỵ✐

(x2 + bk x + ck )

(x − di )

f (x) = an

k=1

di , bk , ck ∈ R✱ 2s + m = n, b2k − 4ck < 0, s, m, n ∈ N∗ ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳

✶✮ ▼å✐ ♥❣❤✐➺♠

x0

❝õ❛ ✤❛ t❤ù❝ ✭✶✳✶✮ ✤➲✉ t❤ä❛ ♠➣♥ ❜➜t

✤➥♥❣ t❤ù❝

|x0 | ≤ 1 +
✷✮ ◆➳✉

am

A
,
|a0 |

A = max |ak |.
1≤k≤n

❧➔ ❤➺ sè ➙♠ ✤➛✉ t✐➯♥ ❝õ❛ ✤❛ t❤ù❝ ✭✶✳✶✮ t❤➻ sè

n

1+

❝➟♥ tr➯♥ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤➣ ❝❤♦✱ tr♦♥❣ ✤â

B

B
am

❧➔

❧➔ ❣✐→ trà

❧ỵ♥ ♥❤➜t ❝õ❛ ♠ỉ✤✉♥ ❝→❝ ❤➺ sè

fn (x) t ữợ fn (x) = g(x)q(x) ✈ỵ✐
deg(g) > 0 ✈➔ deg(q) > 0 t❤➻ t❛ õ g ữợ ừ fn (x) t t g(x)|fn (x)
✳
❤❛② fn (x)✳✳g(x)✳
◆➳✉ g(x)|f (x) ✈➔ g(x)|h(x) t❤➻ t❛ õ g(x) ữợ ừ f (x)
h(x)
tự f (x) h(x) õ ữợ ❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ 0 t❤➻

t❛ ♥â✐ r➡♥❣ ❝❤ó♥❣ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈➔ ✈✐➳t (f (x), h(x)) = 1
tự

ỵ ✤õ ✤➸ ❤❛✐ ✤❛ t❤ù❝ f (x) ✈➔ h(x) ♥❣✉②➯♥ tố
ũ tỗ t tự

u(x)



v(x)

s

f (x)u(x) + h(x)v(x) ≡ 1.

❚➼♥❤ ❝❤➜t ✶✳✶✳
❝→❝ ✤❛ t❤ù❝

g(x)h(x)

◆➳✉ ❝→❝ ✤❛ t❤ù❝

f (x)

h(x)

✈➔

g(x)

♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈➔

♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ t❤➻ ❝→❝ ✤❛ t❤ù❝

f (x)

✈➔

❝ơ♥❣ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✳

❚➼♥❤ ❝❤➜t ✶✳✷✳
f (x)h(x)

✈➔

f (x)

❝❤✐❛ ❤➳t ❝❤♦

❝❤✐❛ ❤➳t ❝❤♦

g(x)✳

f (x), g(x), h(x) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
g(x)✱ g(x) ✈➔ h(x) ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ t❤➻ f (x)

◆➳✉ ❝→❝ ✤❛ t❤ù❝

✼

❚➼♥❤ ❝❤➜t ✶✳✸✳

◆➳✉ ✤❛ t❤ù❝

✈ỵ✐

♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉ t❤➻

g(x)

h(x)

✈➔

❚➼♥❤ ❝❤➜t ✶✳✹✳
m

[f (x)]

✈➔

f (x)

◆➳✉ ❝→❝ ✤❛ t❤ù❝

n

[g(x)]

❝❤✐❛ ❤➳t ❝❤♦ ❝→❝ ✤❛ t❤ù❝

f (x)

f (x)
g(x)

✈➔

❝❤✐❛ ❤➳t ❝❤♦

g(x) ✈➔ h(x)
g(x)h(x).

♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ t

s tố ũ ợ ồ

m, n

ữỡ

ởt số ❜➜t ✤➥♥❣ t❤ù❝ ✤↕✐ sè ❝ì ❜↔♥
❚r♦♥❣ ♣❤➛♥ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ✤❛ tự
số ỡ

ỵ
sỷ

t tự ỳ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ♥❤➙♥✮

x1 , x2 , . . . , xn ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ❑❤✐ ✤â
√
x1 + x2 + · · · + xn
≥ n x1 x2 . . . xn .
n

❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

✭✶✳✹✮

x1 = x2 = . . . = xn .

❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❝â tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉ ❜➡♥❣ t✐➳♥❣ ❱✐➺t ✈➔ ✤÷đ❝ ❣å✐
❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❈æs✐ ✭❈❛✉❝❤②✮✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ữợ
t tự tr õ t t ✏❆▼✲●▼ ■♥❡q✉❛❧✐t②✑✱ ❝❤♦ ♥➯♥ ✈➲
s❛✉✱ t❛ ❣å✐ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❧➔ ✑❇➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉❣ ❜➻♥❤ ❝ë♥❣ ✈➔
tr✉♥❣ ❜➻♥❤ ♥❤➙♥✑✳
❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❦❤→ q✉❡♥ t❤✉ë❝ ✈ỵ✐ ✤❛ sè ❜↕♥ ✤å❝ ✈➔ ✤➣ ✤÷đ❝ ❝❤ù♥❣
♠✐♥❤ tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉ ❜➡♥❣ t✐➳♥❣ ❱✐➺t✱ ♥➯♥ ❝❤ó♥❣ tỉ✐ s➩ ❦❤ỉ♥❣ tr➻♥❤
❜➔② ❝❤ù♥❣ ♠✐♥❤ ♠➔ ❝❤➾ ①➨t ✈➼ ❞ö →♣ ❞ö♥❣✳

❱➼ ❞ö ✶✳✶✳

❈❤♦ ❝→❝ sè ❦❤æ♥❣ ➙♠

x, y, z ✳

❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝

x y z
+ + ≥ x1/2 y 1/3 z 1/6 .
2 3 6
▲í✐ ❣✐↔✐✳

❇➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ t÷ì♥❣ ữỡ ợ

3x + 2y + z

6

6

x3 y 2 z.

t ✈➳ tr→✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ð ❞↕♥❣

3x + 2y + z
x+x+x+y+y+z
=
.
6
6
❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ♥❤➙♥ t❛ ❝â

3x + 2y + z
x+x+x+y+y+z

=
≥
6
6
❇➜t ✤➥♥❣ t❤ù❝ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

6

x3 y 2 z.




ỵ



t tự r

ợ số tỹ tũ ỵ

x1 , x2 , . . . , xn

✈➔

y1 , y2 , . . . , yn

t❛ ❧✉æ♥ ❝â ❜➜t ✤➥♥❣

t❤ù❝

(x21 + x22 + · · · + x2n )(y12 + y22 + · · · + yn2 ) ≥ (x1 y1 + x2 y2 + · · · + xn yn )2 .
✭✶✳✺✮

(x1 , x2 , . . . , xn )
∀i = 1, n.

❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
❤❛✐ ❜ë t✛ ❧➺✱ tự

ỵ

xi = kyi

t tự r

(y1 , y2 , . . . , yn )

✳

x1 , x2 , . . . , xn ✈➔ y1 , y2 , . . . , yn ❧➔ ❤❛✐
∀i = 1, 2, . . . , n✳ ❚❛ ❧✉æ♥ ❝â ❜➜t ✤➥♥❣ t❤ù❝

❈❤♦

❉➜✉ ✤➥♥❣

✈➔

❞➣② sè t❤ü❝✱ tr♦♥❣ ✤â

❧➔

yi > 0,

(x1 + x2 + · · · + xn )2
x2 x2
x2
≤ 1 + 2 + · · · + n.
y1 + y2 + · · · + yn
y 1 y2
yn
x2
xn
x1
=
= ... = .
tự r
y1
y2
yn

ỵ



t tự s

số tỹ tũ ỵ

x1 , x2 , . . . , xn

s❛♦ ❝❤♦

x1 ≤ x2 ≤ . . . ≤ xn ✳

❑❤✐ ✤â

t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿
❛✮ ◆➳✉

y1 ≤ y 2 ≤ . . . ≤ y n

t❤➻

1
(x1 + x2 + . . . + xn )(y1 + y2 + . . . + yn ).
n

x 1 y1 + x 2 y 2 + · · · + x n yn ≥
❜✮ ◆➳✉

y 1 ≥ y2 ≥ . . . ≥ yn

t❤➻

x 1 y1 + x 2 y 2 + · · · + x n yn ≤
❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔

1

(x1 + x2 + . . . + xn )(y1 + y2 + . . . + yn ).
n
❝❤➾ ❦❤✐ x1 = x2 = . . . = xn ❤♦➦❝

y1 = y2 = . . . = yn

ỵ

t tự ❏❡♥s❡♥✮

●✐↔ sû ❤➔♠ sè

❧✐➯♥ tö❝ tr➯♥

I(a, b)✱ tr♦♥❣ ✤â I(a, b) ✤÷đ❝ ♥❣➛♠ ❤✐➸✉
❧➔ ♠ët tr♦♥❣ sè ❝→❝ t➟♣ [a, b], [a, b), (a, b], (a, b)✳ ❑❤✐ ✤â ✤✐➲✉
ừ số f (x) ỗ tr I(a, b) ❧➔
f

f (x)

✳

x1 + x2
2

≤

f (x1 ) + f (x2 )
,

2

∀x1 , x2 ∈ I(a, b).

✶✳✷ ✣❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ♠ët sè ❤➺ t❤ù❝ ❝ì ❜↔♥
▼ư❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❤➺ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ❜➟❝ ❜❛✳


✾

✶✳✷✳✶ ❈ỉ♥❣ t❤ù❝ ❱✐➧t❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ✸
▼➦❝ ❞ị ❝→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ tê♥❣ q✉→t ❦❤ỉ♥❣ ✤÷đ❝ ợ t
ờ tổ ữ t q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❧↕✐
t❤÷í♥❣ ❣➦♣ tr♦♥❣ ❝→❝ ❦➻ t❤✐ ✈➔♦ ✣↕✐ ❤å❝ ✈➔ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✳ ❚r♦♥❣ ♠ö❝
♥➔② tr➻♥❤ ❜➔② ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ổ tự t ừ tự


ỵ


x1 , x2 , x3

✳

✭❈ỉ♥❣ t❤ù❝ ❱✐➧t❡✮

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

ax3 + bx2 + cx + d = 0 (a = 0),
t❤➻




σ1 := x1 + x2 + x3



σ2 := x1 x2 + x1 x3 + x2 x3




σ3 := x1 x2 x3
ự

b
= ,
a
c
= ,
a
d
= .
a

õ ỗ t tự

a(x x1 )(x − x2 )(x − x3 ) ≡ ax3 + bx2 + cx + d
⇔ ax3 − (x1 + x2 + x3 )ax2 + a(x1 x2 + x1 x3 + x2 x3 )x − ax1 x2 x3

≡ ax3 + bx2 + cx + d.
❙♦ s→♥❤ ❤➺ sè ❝→❝ ❧ô② t❤ø❛ ❝ò♥❣ ❜➟❝ ❝õ❛

x

ð ❤❛✐ ✈➳ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥✱

s✉② r ự

ỵ t ữỡ tr ❜➟❝ ❜❛
x3 + ax2 + bx + c = 0

✭✶✳✻✮

✈ỵ✐ ❝→❝ ❤➺ sè ❧➔ ❝→❝ sè t❤ü❝ ✈➔

= −4a3 c + a2 b2 + 18abc − 4b3 − 27c2

✭✶✳✼✮

✤÷đ❝ ❣å✐ ❧➔ ❜✐➺t t❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ❑❤✐ ✤â✿
❛✮ ◆➳✉

> 0✱

t❤➻ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠

x1 , x2 , x3

❧➔ ❝→❝ sè t❤ü❝ ✈➔ ❦❤→❝

♥❤❛✉✳
❜✮ ◆➳✉

< 0✱ t❤➻ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ t❤ü❝✱ ❝á♥ ❤❛✐ ♥❣❤✐➺♠

❦✐❛ ❧➔ ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝ò♥❣ ♥❤❛✉✳
❝✮ ◆➳✉
= 0 ✈➔ a2 − 3b = 0✱ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ❝â ❜❛ ♥❣❤✐➺♠ t❤ü❝✱
tr♦♥❣ ✤â ❝â ❤❛✐ ♥❣❤✐➺♠ trò♥❣ ♥❤❛✉ ✭♥❣❤✐➺♠ ❦➨♣✮✱ ♥❣❤✐➺♠ ❝á♥ ❧↕✐ ❦❤→❝ ❤❛✐


✶✵
♥❣❤✐➺♠ tr➯♥✳ ◆➳✉

=0

a2 − 3b = 0

✈➔

t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❜❛ ♥❣❤✐➺♠

t❤ü❝ ❝ò♥❣ ♥❤❛✉ ✭♥❣❤✐➺♠ ❜ë✐✮✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû

x1 , x2 , x3

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ✭❝â

t❤➸ ❧➔ ❝→❝ sè ♣❤ù❝✱ ♥❤÷♥❣ ➼t ♥❤➜t ❝â ♠ët ♥❣❤✐➺♠ ❧➔ t❤ü❝✮✳ ❑❤✐ ✤â t❤❡♦
❝ỉ♥❣ t❤ù❝ ❱✐➧t❡ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛✱ t❛ ❝â

σ1 = x1 + x2 + x3 = −a, σ2 = x1 x2 + x1 x3 + x2 x3 = b, σ3 = x1 x2 x3 = −c.
❳➨t ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ✤❛ t❤ù❝ ♣❤↔♥ ✤è✐ ①ù♥❣ ✤ì♥ ❣✐↔♥ ♥❤➜t ❝õ❛

x1 , x2 , x3 ✳

= T 2 = (x1 − x2 )2 (x1 − x3 )2 (x2 − x3 )2 .
❚ø ❝→❝ ❤➺ t❤ù❝ ❱✐➧t❡ tr➯♥ ✤➙②✱ t❛ ❝â

= −4a3 c + a2 b2 + 18abc − 4b3 − 27c2 .
❘ã r➔♥❣ ❧➔ ♥➳✉ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ✤➲✉ ❧➔ t❤ü❝ ✈➔ ❦❤→❝ ♥❤❛✉ t❤➻
t❤ü❝ ✈➔ ❦❤→❝ ❦❤ỉ♥❣✱ ❞♦ ✤â

2

= T > 0✳

T

❧➔ sè

✣✐➲✉ ♥❣÷đ❝ ❧↕✐ ✤÷đ❝ s r tứ

ữợ
sỷ

x1

tỹ ỏ

x2 = α + iβ

✈➔

x3 = α − iβ ✳

x2 , x3

❧➔ ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝â ❞↕♥❣✿

❑❤✐ ✤â✱ t❛ ❝â

T = (x1 − α − iβ)(x1 − α + iβ)2iβ = 2iβ[(x1 − α)2 + β 2 ].
❉♦ ✤â

= T 2 = −4β 2 [(x1 − α)2 + β 2 ] < 0.
❝✮ ❚ø ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ♣❤➛♥ ❜✮ t❛ t❤➜② ♥➳✉ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮
❝â ❤❛✐ ♥❣❤✐➺♠ ❜➡♥❣ ♥❤❛✉ t❤➻ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➲✉ ❧➔ t❤ü❝

= 0✳

✈➔

✣➸ ❧➔♠ s→♥❣ tä ❦❤✐ ♥➔♦ ❝❤➾ ❝â ❤❛✐ ♥❣❤✐➺♠ ❜➡♥❣ ♥❤❛✉ ✭♥❣❤✐➺♠

❦➨♣✮✱ ❤♦➦❝ ❝↔ ❜❛ ♥❣❤✐➺♠ ❜➡♥❣ ♥❤❛✉ ✭♥❣❤✐➺♠ ❜ë✐✮✱ t❛ ①➨t ❜✐➸✉ t❤ù❝

1

= (x1 − x2 )2 + (x2 − x3 )2 + (x3 − x1 )2 = 2(σ12 − 3σ2 ) = 2(a2 − 3b).

❘ã r➔♥❣ ♥➳✉

x1 , x2 , x3

❧➔ ❝→❝ sè t❤ü❝ t❤➻

1

= 0✱

tù❝ ❧➔

a2 = 3b

❦❤✐ ✈➔

❝❤➾ ❦❤✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ❝â ❜❛ ♥❣❤✐➺♠ t❤ü❝ ❜➡♥❣ ♥❤❛✉ ✭♥❣❤✐➺♠ ❜ë✐✮✳

= 0 ✈➔ a2 = 3b t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ❝â ♥❣❤✐➺♠ ❜ë✐✱ ❝á♥ ♥➳✉
= 0 ✈➔ a2 = 3b t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ sè ❦➨♣✳ ✣à♥❤ ỵ ữủ ự





ử

ởt ữỡ tr ❜➟❝ ❜❛ ❝â ❝→❝ ♥❣❤✐➺♠ ❧➔ ❜➻♥❤

♣❤÷ì♥❣ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

u3 − 2u2 + u − 12 = 0.


✶✶
▲í✐ ❣✐↔✐✳

r1 , r2 , r3

❑➼ ❤✐➺✉

u1 , u2 , u3

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ✈➔

❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ sì ❝➜♣ ❝õ❛ ❝→❝ ❜✐➳♥

u1 , u2 , u3

ỵ t t õ

r1 = u1 + u2 + u3 = 2,

r2 = u1 u2 + u2 u3 + u3 u1 = 1,

r3 = u1 u2 u3 = 12.

●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ ❝➛♥ ❧➟♣ ❝â ❞↕♥❣

x 3 − σ1 x 2 + σ2 x − σ3 = 0
✈➔

x1 , x2 , x3

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♥â✳ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤➲ ❜➔✐ t❛ ❝â

σ1 = x1 + x2 + x3 = u21 + u22 + u23 = r12 − 2r2 = 22 − 2 = 2,
σ2 = x1 x2 + x2 x3 + x3 x1 = u21 u22 + u22 u23 + u23 u21
= r22 − 2r1 r3 = 1 − 2.2.12 = −47.
σ3 = x1 x2 x3 = u21 u22 u23 = r32 = 122 = 144.
❱➟② ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝➛♥ ❧➟♣ s➩ ❧➔

x3 − 2x2 − 47x − 144 = 0.

❱➼ ❞ö ✶✳✸✳

❈❤♦

x1 , x2 , x3

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

ax3 − ax2 + bx + b = 0,

(a.b = 0).

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

(x1 + x2 + x3 )
▲í✐ ❣✐↔✐✳

1
1
1
+
+
x1 x2 x3

= −1.

❚❤❡♦ ỵ t t õ

b
b
1 = x1 + x2 + x3 = 1, σ2 = x1 x2 + x2 x3 + x3 x1 = , σ3 = x1 x2 x3 = − .
a
a
❑❤✐ ✤â

1
1
x1 x2 + x2 x3 + x3 x1
σ2
1
+
+

=
=
= −1.
x1 x2 x3
x1 x2 x3
σ3
❉♦ ✤â t❛ ❝â

❱➼ ❞ö ✶✳✹✳

❚➻♠

(x1 + x2 + x3 )

1
1
1
+
+
x1 x2 x3

a

x1 , x2 , x3

✤➸ ❝→❝ ♥❣❤✐➺♠

= −1.

❝õ❛ ✤❛ t❤ù❝

f (x) = x3 − 6x2 + ax + a


✶✷
t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝

(x1 − 3)2 + (x2 − 3)2 + (x3 − 3)2 = 0.
▲í✐ ❣✐↔✐✳

y1 , y2 , y3

✣➦t

y = x − 3.

❇➔✐ t♦→♥ trð t❤➔♥❤✿ ❚➻♠

a

✤➸ ❝→❝ ♥❣❤✐➺♠

❝õ❛ ✤❛ t❤ù❝

g(y) = f (y + 3) = (y + 3)3 − 6(y + 3)2 + a(y + 3) + a
= y 3 + 3y 2 + (a − 9)y + 4a − 27
t❤ä❛ ♠➣♥ ❤➺ t❤ù❝

y13 + y23 + y33 = 0.
ỵ t t õ

1 = y1 + y2 + y3 = −3,

σ2 = y1 y2 + y2 y3 + y3 y1 = a − 9,

σ3 = y1 y2 y3 = 27 − 4a.
❑❤✐ ✤â

y13 + y23 + y33 = σ13 − 3σ1 σ2 + 3σ3
= (−3)3 − 3(−3)(a − 9) + 3(27 − 4a)
= −27 − 3a.
❉♦ ✤â✱ t❛ ❝â

y13 + y23 + y33 = 0 ⇔ −27 − 3a = 0 ⇔ a = −9.
❱➟② ❣✐→ trà ❝➛♥ t➻♠ ❝õ❛

❱➼ ❞ö ✶✳✺✳

❇✐➳t r➡♥❣

a

❧➔

t, u, v

a = −9✳
❧➔ ❜❛ ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

x3 + ax2 + bx + c = 0,

tr♦♥❣ ✤â

✭✶✳✽✮

a, b, c ❧➔ ❝→❝ sè t❤ü❝✳ ❚➻♠ ✤✐➲✉ ❦✐➺♥ ❝õ❛ a, b, c ✤➸ t3 , u3 , v 3

♥❣❤✐➺♠

✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤

x3 + a3 x2 + b3 x + c3 = 0.
▲í✐ ❣✐↔✐✳

✭✶✳✾✮

⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❱✐➧t❡ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✽✮✱ t❛ ❝â

σ1 = t + u + v = −a,

σ2 = tu + uv + vt = b,

σ3 = tuv = −c.

✭✶✳✶✵✮


✶✸

t3 , u3 , v 3

●✐↔ sû

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮✳ ❚❤❡♦ ❝ỉ♥❣ t❤ù❝

❱✐➧t❡✱ t❛ ❝â




t3 + u3 + v 3 = −a3 ,


t3 u3 + u3 v 3 + v 3 t3 = b3 ,



t3 u3 v 3 = −c3 .
❚❤❛② ❝→❝ ❣✐→ trà ❝õ❛
❤➺ t❤ù❝

c = ab✳

❱ỵ✐

⇔




σ13 − 3σ1 σ2 + 3σ3 = −a3 ,




σ23 − 3σ1 σ2 σ3 + 3σ32 = b3 ,



σ = −c3 .
3

σ1 , σ2 , σ3 tø ✭✶✳✶✵✮ ✈➔♦ ❤➺ tr➯♥ ✈➔ rót ❣å♥✱ t❛ t❤✉ ✤÷đ❝
c = ab✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✽✮ trð t❤➔♥❤

x3 + ax2 + bx + ab = 0 ⇔ (x + a)(x2 + b) = 0.
❱➻ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ t❤ü❝✱ ♥➯♥ t❛ ♣❤↔✐ ❝â

b ≤ 0✳

❱➟②✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝õ❛

a, b, c

❧➔

c = ab

✈➔

b ≤ 0✳

✶✳✷✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜❛ ➞♥
●✐↔ sû

P (x, y, z), Q(x, y, z), R(x, y, z) ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ❳➨t ❤➺

♣❤÷ì♥❣ tr➻♥❤




P (x, y, z) = 0,


Q(x, y, z) = 0,



R(x, y, z) = 0.

✭✶✳✶✶✮

❇➡♥❣ ❝→❝❤ ✤➦t

x + y + z = σ1 ,

σ2 = xy + yz + zx,

σ3 = xyz,

t❛ ✤÷❛ ❤➺ ✭✶✳✶✶✮ ✈➲ ❞↕♥❣




p(σ , σ , σ ) = 0,

 1 2 3

q(σ1 , σ2 , σ3 ) = 0,



r(σ , σ , σ ) = 0.
1 2 3

✭✶✳✶✷✮

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✷✮ t❤÷í♥❣ ✤ì♥ ❣✐↔♥ ❤ì♥ ❤➺ ✭✶✳✶✶✮ ✈➔ t❛ ❝â t❤➸ ❞➵

σ1 , σ2 , σ3 ✳ ❙❛✉ ❦❤✐ t➻♠ ✤÷đ❝ ❝→❝ ❣✐→ trà ❝õ❛ σ1 , σ2 , σ3 ✱
trà ❝õ❛ ❝→❝ ➞♥ sè x, y, z ✳ ✣✐➲✉ ♥➔② ❞➵ ❞➔♥❣ t❤ü❝ ❤✐➺♥

❞➔♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠
❝➛♥ ♣❤↔✐ t➻♠ ❝→❝ ❣✐→

✤÷đ❝ ♥❤í ✤à♥❤ ❧➼ s❛✉ ✤➙②✳

ỵ



sỷ 1, 2, 3 ❝→❝ sè t❤ü❝ ♥➔♦ ✤â✳ ❑❤✐ ✤â

♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛

u3 − σ1 u2 + σ2 u − σ3 = 0

✭✶✳✶✸✮




x+y+z
= σ1 ,


xy + yz + zx = σ2 ,



xyz
= σ3 .



ữỡ tr

ợ ữ s

u1 , u2 , u3

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✭✶✳✶✸✮✱ t❤➻ ❤➺ ✭✶✳✶✹✮ ❝â ❝→❝ ♥❣❤✐➺♠




x = u1 ,

 1

y1 = u2 ,



z = u ;
1
3



x = u2 ,

 4
y4 = u3 ,




z = u ;
4

1




x = u1 ,

 2
y2 = u3 ,



z = u ;
2
2



x = u3 ,

 5
y5 = u1 ,




z = u ;
5

2




x = u2 ,

 3
y3 = u1 ,



z = u ;
3
3



x = u3 ,

 6
y6 = u2 ,



z = u ;
6

1

✈➔ ♥❣♦➔✐ r❛ ❦❤æ♥❣ ❝á♥ ❝→❝ ♥❣❤✐➺♠ ♥➔♦ ❦❤→❝✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉

z = c

❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✹✮✱ t❤➻ ❝→❝ sè

a, b, c

x = a, y = b,

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣

tr➻♥❤ ✭✶✳✶✸✮✳

❈❤ù♥❣ ♠✐♥❤✳

●✐↔ sû

u1 , u2 , u3

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮✳

❑❤✐ ✤â t❛ õ ỗ t tự

u3 1 u2 + 2 u − σ3 = (u − u1 )(u − u2 )(u − u3 ).
❚ø ✤â t❛ ❝â ❝→❝ ❤➺ t❤ù❝ ❱✐➧t❡✿




u + u2 + u3

 1

u1 u2 + u1 u3 + u2 u3



u u u
1 2 3
❙✉② r❛

u1 , u2 , u3

= σ1 ,
= σ2 ,
= σ3 .

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✹✮✳ ◆❣♦➔✐ r❛ ❝á♥ ♥➠♠ ♥❣❤✐➺♠

♥ú❛ ♥❤➟♥ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ❤♦→♥ ✈à ❝→❝ ❣✐→ trà ❝õ❛ ❝→❝ ➞♥ sè✳ ❱➜♥ ✤➲ ❤➺


✶✺
✭✶✳✶✹✮ ❦❤ỉ♥❣ ❝á♥ ♥❣❤✐➺♠ ♥➔♦ ❦❤→❝ s➩ ✤÷đ❝ ❧➔♠ s→♥❣ tọ ữợ
sỷ

x = a, y = b, z = c ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✹✮✱



a+b+c
= σ1 ,



♥❣❤➽❛ ❧➔

ab + bc + ac = σ2 ,



abc
= σ3 .

❑❤✐ ✤â t❛ ❝â

u3 − σ2 u2 + σ2 u − σ3 = u3 − (a + b + c)u2 + (ab + bc + ca)u − abc
= (u − a)(u − b)(u − c).
✣✐➲✉ ✤â ❝❤ù♥❣ tä r➡♥❣ ❝→❝ sè

a, b, c

ừ ữỡ tr

ỵ ữủ ự

ỵ sỷ 1, 2, 3 ❧➔ ❝→❝ sè t❤ü❝ ✤➣ ❝❤♦✳ ✣➸ ❝→❝ sè x, y, z
①→❝ ✤à♥❤ ❜ð✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✹✮ ❧➔ ❝→❝ sè t❤ü❝✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❧➔

= −4σ13 σ3 + σ12 σ22 + 18σ1 σ2 σ3 − 4σ23 − 27σ3 ≥ 0.
◆❣♦➔✐ r❛✱ ✤➸ ❝→❝ sè

x, y, z

❧➔ ❦❤æ♥❣ t

1 0,
ự
ỵ

x, y, z

sỷ



x, y, z

σ2 ≥ 0,

σ3 ≥ 0.

❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✹✮✳ ❑❤✐ ✤â t❤❡♦ ✣à♥❤

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮✳ ❚❤❡♦ ỵ

ữỡ tr õ tỹ ❝❤➾ ❦❤✐ ❜✐➺t t❤ù❝ ❝õ❛ ♥â ❦❤æ♥❣

x, y, z ❧➔ ❦❤æ♥❣ ➙♠✱
t❤➻ ❤✐➸♥ ♥❤✐➯♥ σi ≥ 0
(i = 1, 2, 3)✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ σi ≥ 0 (i = 1, 2, 3) ✈➔
➙♠✱ ♥❣❤➽❛ ❧➔ ✭✶✳✶✺✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ◆❣♦➔✐ r❛✱ ♥➳✉ ❝→❝ sè

✭✶✳✶✺✮ ✤÷đ❝ t❤ä❛ ♠➣♥✱ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮ ❦❤æ♥❣ t❤➸ ❝â ♥❣❤✐➺♠ ➙♠✳
❚❤➟t ✈➟②✱ tr♦♥❣ ✭✶✳✶✸✮ t❤❛②

u = −v

t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤

v 3 + σ1 v 2 + σ2 v + σ2 = 0.
❱➻

σi ≥ 0 (i = 1, 2, 3)✱

✭✶✳✶✻✮

♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✻✮ ❦❤ỉ♥❣ t❤➸ ❝â ♥❣❤✐➺♠

❞÷ì♥❣✱ ❞♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮ ❦❤ỉ♥❣ t❤➸ ❝â ♥❣❤✐➺♠ ➙♠✳ ❚ø ✤â s✉② r❛

x, y, z

❧➔ ❝→❝ sè ❦❤æ♥❣ ỵ ữủ ự

ử

ữỡ tr➻♥❤




x + y + z = 2,


x2 + y 2 + z 2 = 6,



x3 + y 3 + z 3 = 8.


✶✻
▲í✐ ❣✐↔✐✳

✣➦t

x + y + z = σ1 , σ2 = xy + yz + zx, σ3 = xyz.

❙û ❞ư♥❣

❝ỉ♥❣ t❤ù❝ ❲❛r✐♥❣ t❛ ❝â

x2 + y 2 + z 2 = σ12 − 2σ2 ,

x3 + y 3 + z 3 = σ13 − 3σ1 σ2 + 3σ3 .

❉♦ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉ trð t❤➔♥❤




σ = 2,

 1
σ12 − 2σ2 = 6,



σ 3 − 3σ σ + 3σ = 8.
1 2
3
1
●✐↔✐ ❤➺ ♥➔② t❛ t➻♠ ✤÷đ❝
t❛ ❝â

x, y, z

σ1 = 2, σ2 = −1, σ3 = −2.

❚❤❡♦ ✣à♥❤ ỵ

ừ ữỡ tr

u3 2u2 u + 2 = 0 ⇔ (u2 − 1)(u − 2) = 0.
u1 = −1, u2 = 1, u3 = 2✳

❜ë (x, y, z) s❛✉ ✤➙②✿

◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➔
♥❣❤✐➺♠ ❝õ❛ ❤➺ ✤➣ ❝❤♦ ❧➔ ♥❤ú♥❣

❚ø ✤â s✉② r❛

(−1, 1, 2), (−1, 2, 1), (1, −1, 2), (1, 2, 1), (2, −1, 1), (2, 1, −1).

✶✳✷✳✸ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû
❚r♦♥❣ ♠ö❝ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✈➔ ♣❤↔♥
✤è✐ ①ù♥❣ ❜❛ ❜✐➳♥ ✈➔♦ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ♥❤➙♥ tû✳
●✐↔ sû

f (x, y, z)

❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛ t

f (x, y, z)

t tỷ trữợ ❤➳t ❝➛♥ ♣❤↔✐ ❜✐➸✉ ❞✐➵♥ ♥â q✉❛ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣

ϕ(σ1 , σ2 , σ3 )✱ s❛✉ ✤â ❝è ❣➢♥❣ ♣❤➙♥ t➼❝❤ ✤❛
t❤ù❝ ❝✉è✐ ❝ò♥❣ t❤➔♥❤ ♥❤➙♥ tû✳ ◆➳✉ tr♦♥❣ ❝→❝ ♥❤➙♥ tû ❝õ❛ f (x, y, z) ❝â ✤❛
t❤ù❝ ❦❤æ♥❣ ✤è✐ ①ù♥❣ h(x, y, z)✱ t❤➻ ❞♦ f (x, y, z) ❧➔ ✤è✐ ①ù♥❣ s➩ ♣❤↔✐ ❝â ❝→❝
♥❤➙♥ tû ♥❤➟♥ ✤÷đ❝ tø h(x, y, z) ❜➡♥❣ ❝→❝❤ ❤♦→♥ ✈à ❝→❝ ❜✐➳♥ x, y, z ♥❣❤➽❛
❝ì sð

σ1 , σ2 , σ3

✤➸ ✤÷đ❝ ✤❛ t❤ù❝

❧➔ ❝â ❝→❝ ♥❤➙♥ tû ❞↕♥❣✿

h(x, y, z), h(x, z, x), h(y, x, z), h(y, z, x), h(z, x, y), h(z, y, x).
◆➳✉ tr♦♥❣ ❝→❝ ♥❤➙♥ tû ❝â ♥❤➙♥ tû
❤❛✐ ❜✐➳♥✱ t❤➼ ❞ư ✤è✐ ✈ỵ✐

x, y

g(x, y, z)

❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝❤➾ ✈ỵ✐

♥❣❤➽❛ ❧➔

g(x, y, z) = g(y, x, z),
t❤➻ ❝→❝ ♥❤➙♥ tû ❝ò♥❣ ❞↕♥❣ s➩ ❧➔

g(x, y, z), g(y, z, x), g(z, x, y).


✶✼
◆➳✉ ♥❤÷ tr♦♥❣ ❝→❝ ♥❤➙♥ tû ❝â ♥❤➙♥ tû

k(x, y, z)

✤è✐ ①ù♥❣ ❝❤➤♥✱ ♥❣❤➽❛ ❧➔

k(x, y, z) = k(y, z, x) = k(z, x, y),

t❤➻ ❝→❝ ♥❤➙♥ tû ❝ò♥❣ ❞↕♥❣ s➩ ❧➔

k(x, y, z), k(y, z, x).
◆❤÷ ✈➟②✱ tr♦♥❣ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ♥❤➙♥ tû ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣

f (x, y, z)

❝â t❤➸ ❣➦♣ ❝→❝ ♥❤➙♥ tû ❞↕♥❣ s❛✉ ✤➙②✿

p(x, y, z)✳
k(x, y, z), k(y, z, x)✱ tr♦♥❣

✶✮ ◆❤➙♥ tû ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
✷✮ ◆❤➙♥ tû ❝â ❞↕♥❣

✤â

k(x, y, z)

❧➔ ✤è✐ ①ù♥❣

❝❤➤♥✳

g(x, y, z), g(y, z, x), g(z, x, y)✱ tr♦♥❣ ✤â g(x, y, z) ✤è✐
①ù♥❣ t❤❡♦ ❤❛✐ ❜✐➳♥✱ t❤➼ ❞ö x, y ✳
✹✮ ◆❤➙♥ tû ❞↕♥❣ h(x, y, z), h(x, z, x), h(y, x, z), h(y, z, x), h(z, x, y),
h(z, y, x)✱ tr♦♥❣ ✤â h(x, y, z) ❦❤ỉ♥❣ ❝â t➼♥❤ ✤è✐ ①ù♥❣✳
✣è✐ ✈ỵ✐ ✤❛ t❤ù❝ ♣❤↔♥ ✤è✐ ①ù♥❣ f (x, y, z)✱ t❛ ❝â ♣❤➙♥ t➼❝❤
✸✮ ◆❤➙♥ tû ❝â ❞↕♥❣

f (x, y, z) = T (x, y, z)g(x, y, z),
tr♦♥❣ ✤â

T (x, y, z)

❧➔ ✤❛ t❤ù❝ ♣❤↔♥ ✤è✐ ①ù♥❣ ✤ì♥ ❣✐↔♥ ♥❤➜t✱ ❝á♥

g(x, y, z)

❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ◆❣♦➔✐ r❛✱ ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t
❝â ❦➳t q✉↔ s❛✉ ✤➙②✳

▼➺♥❤ ✤➲ ✶✳✶✳ ❑➼ ❤✐➺✉ θm(x, y, z) ❧➔ ✤❛ t❤ù❝ ♣❤↔♥ ✤è✐ ①ù♥❣ ❜➟❝ m✳ ❑❤✐
✤â

θ3 (x, y, z) = aT (x, y, z)✱
θ4 (x, y, z) = aT (x, y, z)σ1 ✱
θ5 (x, y, z) = T (x, y, z)(aσ12 + bσ2 ),
θ6 (x, y, z) = T (x, y, z)(aσ13 + bσ1 σ2 + cσ3 )✱

tr♦♥❣ ✤â

a, b, c

❧➔ ❝→❝ ❤➡♥❣

sè✳
❚❛ ①➨t ❝→❝ ✈➼ ❞ö s❛✉ ✤➙②✳

❱➼ ❞ö ✶✳✼✳

P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ s❛✉ t❤➔♥❤ ♥❤➙♥ tû

f (x, y, z) = x3 + y 3 + z 3 − 3xyz.
▲í✐ ❣✐↔✐✳

❚❛ ❝â

f (x, y, z) = (σ13 − 3σ1 σ2 + 3σ3 ) − 3σ3 = σ13 − 3σ1 σ2
= σ1 (σ12 − 3σ2 ) = (x + y + z)[(x + y + z)2 − 3(xy + yz + zx)]
= (x + y + z)(x2 + y 2 + z 2 − xy − yz − zx).


✶✽

❱➼ ❞ö ✶✳✽✳

P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ s❛✉ t❤➔♥❤ ♥❤➙♥ tû

f (x, y, z) = 2x2 y 2 + 2x2 z 2 + 2y 2 z 2 − x4 − y 4 − z 4 .
▲í✐ ❣✐↔✐✳

❚❛ ❝â

f (x, y, z) = 2x2 y 2 + 2x2 z 2 + 2y 2 z 2 − x4 − y 4 − z 4
= 2(σ22 − 2σ1 σ3 ) − (σ14 − 4σ12 σ2 + 2σ22 + 4σ1 σ3 )
= −σ14 + 4σ12 σ2 − 8σ1 σ3
= σ1 (4σ1 σ2 − σ13 − 8σ3 ).
σ1 = x + y + z ✳ ◆❤÷♥❣ ✈➻ ✤❛ t❤ù❝
✤➣ ❝❤♦ ❧➔ ❤➔♠ ❝❤➤♥ ✤è✐ ✈ỵ✐ x, y, z ✱ ♥➯♥ ♥â ❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ −x + y + z,

x − y + z, x + y − z ✳ ❈ô♥❣ ✈➻ ✤❛ t❤ù❝ ✤➣ ❝❤♦ ❝â ❜➟❝ ❜➡♥❣ 4✱ ♥➯♥ t❛ ❝â
◆❤÷ ✈➟②✱ ✤❛ t❤ù❝ ✤➣ ❝❤♦ ❝❤✐❛ ❤➳t ❝❤♦

f (x, y, z) = C(x + y + z)(−x + y + z)(x − y + z)(x + y − z),
C ❧➔ ❤➡♥❣ sè ♥➔♦ ✤â✳ ✣➸ ①→❝
✤÷đ❝ C = 1✳ ❱➟② t❛ ❝â ❦➳t q✉↔✿

tr♦♥❣ ✤â
t➻♠

✤à♥❤

C

t❛ ❝❤♦

x=y=z =1

✈➔

2x2 y 2 + 2x2 z 2 + 2y 2 z 2 − x4 − y 4 − z 4
=(x + y + z)(−x + y + z)(x − y + z)(x + y − z).

✶✳✷✳✹ ❚➼♥❤ ❝❤✐❛ ❤➳t ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❚r♦♥❣ ♠ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ✈➼ ❞ö ✈➲ t➼♥❤ ❝❤✐❛ ❤➳t ❝õ❛ ❝→❝ ✤❛ t❤ù❝
✤è✐ ①ù♥❣✳

❱➼ ❞ư ✶✳✾✳

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ ❞÷ì♥❣

n✱

✤❛ t❤ù❝

f (x, y, z) = (x + y + z)2n+1 − x2n+1 − y 2n+1 − z 2n+1
❝❤✐❛ ❤➳t ❝❤♦ ✤❛ t❤ù❝

g(x, y, z) = (x + y + z)3 − x3 − y 3 − z 3 .
g(x, y, z) t❤➔♥❤ ♥❤➙♥ tû✳ ❱➻ ❦❤✐ x = −y,
x = −z, y = −z t❤➻ g = 0✱ ♥➯♥ t❤❡♦ ✤à♥❤ ❧➼ ❇❡③♦✉t ✤❛ t❤ù❝ g(x, y, z) ❝❤✐❛
❤➳t ❝❤♦ (x + y)(x + z)(y + z)✳ ▼➦t ❦❤→❝✱ ✈➻ ❜➟❝ ❝õ❛ g ❜➡♥❣ 3✱ ♥➯♥ õ õ

ớ rữợ t t t



g(x, y, z) = a(x + y)(x + z)(y + z).
❈❤♦

x=y=z=1

t❛ t➻♠ ✤÷đ❝

a = 3✳

❱➙② t❛ ❝â

g(x, y, z) = (x + y + z)3 − x3 − y 3 − z 3 = 3(x + y)(x + z)(y + z).

✶✾

f (x, y, z) ❝❤✐❛ ❤➳t ❝❤♦ (x + y)(x + z)(y + z)
tù❝ ❧➔ f (x, y, z) ❝❤✐❛ t g(x, y, z).

tữỡ tỹ t t
ợ ♠å✐

n

♥❣✉②➯♥ ❞÷ì♥❣✱

❱➼ ❞ư ✶✳✶✵✳
❝❤♦

x−y

❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ♥➳✉ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣

t❤➻ ♥â ❝❤✐❛ ❤➳t ❝❤♦

▲í✐ ❣✐↔✐✳

f (x, y, z)

❝❤✐❛ ❤➳t

(x − y)2 (x − z)2 (y − z)2 .

●✐↔ sû r➡♥❣

f (x, y, z) = (x − y)g(x, y, z).
❱➻

f (y, x, z) = (y − x)g(y, x, z) = −(x − y)g(y, x, z),
♥➯♥

g(y, x, z) = −g(x, y, z),
g(x, y, z) ❧➔ ✤❛ t❤ù❝ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤❡♦ ❤❛✐ ❜✐➳♥ x, y ✳ ❱➟② g(x, y, z)
2
❝❤✐❛ ❤➳t ❝❤♦ x − y ✳ ❉♦ ✤â f (x, y, z) ❝❤✐❛ ❤➳t ❝❤♦ (x − y) ✳ ❱➻ f (x, y, z) ❧➔
✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ♥➯♥ ✈❛✐ trá ❝õ❛ x, y, z ❧➔ ♥❤÷ ♥❤❛✉✱ ❝❤♦ ♥➯♥ f (x, y, z)
2
2
❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ (x − z) ✈➔ (y − z) ✳ ❱➟② f (x, y, z) ❝❤✐❛ ❤➳t ❝❤♦
(x − y)2 (x − z)2 (y − z)2 .

s✉② r❛

❱➼ ❞ö ✶✳✶✶✳
❤➳t ❝❤♦

❚➻♠ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ✤❛ t❤ù❝

x3 + y 3 + z 3 + kxyz

❝❤✐❛

x + y + z✳

▲í✐ ❣✐↔✐✳

❳➨t ✤❛ t❤ù❝ t❤❡♦ ❜✐➳♥

x

f (x) = x3 + (kyz)x + (y 3 + z 3 ).
f (x) ❝❤✐❛
f (−y − z) = 0✳ ❚❛ õ

ỵ t


t

x + y + z = x − (−y − z)

❦❤✐

f (−y − z) = −(y + z)3 − kyz(y + z) + (y 3 + z 3 )
= (k + 3)yz(y + z) = 0, ∀y, z.
❚ø ✤â s✉② r❛

k = −3.

❱➟② ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸
❧➔

x3 + y 3 + z 3 + kxyz

❝❤✐❛ ❤➳t ❝❤♦

x+y+z

k = −3.

✶✳✸ ✣❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝
P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ tê♥❣ q✉→t ❝â ❞↕♥❣

x3 + ax2 + bx + c = 0.

✭✶✳✶✼✮


✷✵

❞↕♥❣

a
✳
3

x=y−

❑❤✐ ✤â ❜➡♥❣ ❝→❝❤ ✤➦t

a 2
a
+b y−

+c=0
3
3
⇔ y 3 − py = q.
a2
2a3 ab
❱ỵ✐ p =
− b, q = −
+
− c.
3
27
3
✶✮ ◆➳✉ p = 0 ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳
p
✷✮ ◆➳✉ p > 0✳ ✣➦t y = 2
t✳ ❑❤✐ õ t ữủ ữỡ tr
3

3
3q
4t3 3t = m ợ m = √ .
2p p
y−

❛✮ ◆➳✉

❜✮ ◆➳✉

|m| ≤ 1✱

a
3

P❤÷ì♥❣ tr➻♥❤ õ t t ữợ

3

t

+a y

m = cos

ữỡ tr➻♥❤ ❝â ❜❛ ♥❣❤✐➺♠

α ± 2π
α
.
t = cos ; t = cos
3
3
√
1 3
1
|m| > 1✱ ✤➦t m =
d + 3 ✱ tr♦♥❣ ✤â d3 = m ± m2 − 1✳
2
d

❑❤✐ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t

t=
✸✮ ◆➳✉

1
1
d+
2
d
p < 0✱

=

✤➦t

1
2

y=2

3

m2 − 1 +

m+

−p
t✳
3

m=

1 3
1
d − 3
2
d

m2 − 1 .

m−

❑❤✐ ✤â✱ t ữủ ữỡ tr

4t3 + 3t = m
t

3

ợ

tr õ


3 3q
m= √ .
2p p

d3 = m ±

√

m2 + 1✳

❑❤✐ ✤â ♣❤÷ì♥❣

tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t

t=

1
1
d+
2
d

=

1
2

3

m+

√

m2 + 1 +

3

m−

❚❛ ❝â ❝→❝ ỵ ỡ tr t s

ỵ

ỵ số

r t

ABC



sin

t ổ õ

a
b
c
=
=
= 2R.
sin A sin B
sin C

√

m2 + 1 .




ỵ

ỵ số s



r t t❛ ❧✉æ♥ ❝â

a2 = b2 + c2 − 2bc cos A;
b2 = a2 + c2 − 2ac cos B;
c2 = a2 + b2 − 2ab cos C.

❇➔✐ t♦→♥ ✶✳✶✳

ABC

✣ë ❞➔✐ ❜❛ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝

✭❣✐↔ sû ❧➛♥ ❧÷đt ❧➔

a, b, c✮

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

t3 − 2pt2 + (p2 + r2 + 4Rr)t − 4pRr = 0.

a, b, c ❧➔ ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝ ❆❇❈✱ ❦❤✐

ự sỷ
t ỵ số



sin

ổ tự ❣â❝ ♥❤➙♥ ✤æ✐ t❛ ❝â

a = 2R sin A = 4R sin

A
A
cos .
2
2

✭✶✳✶✾✮

❚❤❡♦ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ t❤➻

A
A
2.
p − a = r cot = r
A

2
sin
2
cos

✭✶✳✷✵✮

▲➜② ✭✶✳✶✾✮ ❝❤✐❛ ❝❤♦ ✭✶✳✷✵✮✱ t❛ ✤÷đ❝

sin2

A
ar
=
.
2
4R(p − a)

▲➜② ✭✶✳✶✾✮ ♥❤➙♥ ợ t ữủ

cos2

A a(p a)
=
.
2
4Rr

ứ õ s r

ar
a(p a)
A
A
+
= sin2 + cos2 = 1.
4R(p a)
4Rr
2
2
ỗ ♠➝✉ sè t❛ ✤÷đ❝

r2 a + a(p − a)2 = 4Rr(p − a)
⇔ a3 − 2pa2 + (p2 + r2 + 4Rr)a − 4pRr = 0.
❱➟②

a

❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✽✮✳ ❚÷ì♥❣ tü

❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳

b, c

❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✽✮✳ ❚❛


✷✷

❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤

❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t❛ q✉❛♥ t➙♠ ❝❤õ ②➳✉ ✤➳♥ ❝→❝ ❞↕♥❣ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
✈➔ ✤❛ t❤ù❝ ỗ số tỹ tr tỹ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤
❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø t➔✐ ❧✐➺✉ ❬✸❪✳

✷✳✶ ❇➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ✤❛ t❤ù❝ ❜➟❝ ❜❛
✷✳✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳

▼ët ✤ì♥ t❤ù❝

ϕ(x, y, z)

❝õ❛ ❝→❝ ❜✐➳♥

x, y, z

✤÷đ❝ ❤✐➸✉

❧➔ ❤➔♠ sè ❝â ❞↕♥❣

ϕ(x, y, z) = aklm xk y l z m ,
k, l, m ∈ N ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ❝→❝ ❜✐➳♥ x, y, z ❀ sè aklm ∈ R∗ ✤÷đ❝
❣å✐ ❧➔ ❤➺ sè ❝õ❛ ✤ì♥ t❤ù❝✱ ❝á♥ sè k + j + m ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ✤ì♥ t❤ù❝
ϕ(x, y, z)✳
tr♦♥❣ ✤â

✣à♥❤ ♥❣❤➽❛ ✷✳✷✳

▼ët ❤➔♠ sè

P (x, y, z)

❝õ❛ ❝→❝ ❜✐➳♥

x, y, z

✤÷đ❝ ❣å✐ ❧➔

♠ët ✤❛ t❤ù❝✱ ♥➳✉ ♥â ❝â t❤➸ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ð ❞↕♥❣ tê♥❣ ❤ú✉ ❤↕♥ ❝→❝ ✤ì♥
t❤ù❝✿

aklm xk y l z m

P (x, y, z) =
k+l+m≤n

❇➟❝ ❧ỵ♥ ♥❤➜t ❝õ❛ ❝→❝ ✤ì♥ t❤ù❝ tr♦♥❣ ✤❛ t❤ù❝ ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝✳

✣à♥❤ ♥❣❤➽❛ ✷✳✸✳

P (x, y, z) ✤÷đ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ♥➳✉ ♥â
❤♦→♥ ✈à ❝õ❛ x, y, z ♥❣❤➽❛ ❧➔

✣❛ t❤ù❝

❦❤æ♥❣ t❤❛② ✤ê✐ ✈ỵ✐ ♠å✐

P (x, y, z) = P (y, x, z) = P (z, y, x) = P (x, z, y) = P (y, z, x) = P (z, x, y).

✣à♥❤ ♥❣❤➽❛ ✷✳✹✳

✣❛ t❤ù❝ ♣❤↔♥ ✤è✐ ①ù♥❣ ❧➔ ✤❛ t❤ù❝ t❤❛② ✤ê✐ ❞➜✉ ❦❤✐ t❤❛②

✤ê✐ ✈à tr➼ ❝õ❛ ❤❛✐ ❜✐➳♥ ❜➜t ❦ý✳