✣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✿ ✻✵ ✹✻ ✵✶ ✶✸
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ❱➠♥ ▼➟✉
❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✾
✐
▼ö❝ ❧ö❝
▼Ð ✣❺❯
❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥
✶
✸
✶✳✶
▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ê ✤✐➸♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝
✶✳✷
✣❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ♠ët sè ❤➺ t❤ù❝ ❝ì ❜↔♥
✶✳✸
✳ ✳ ✳ ✳
✸
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✽
✶✳✷✳✶
❈æ♥❣ 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 ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✷
✷✳✶✳✷
❈→❝ ✤à♥❤ ❧þ ❝ì ❜↔♥ ❝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❖
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✽
✹✼
✹✽
t tự õ trỏ rt q trồ tr ồ ờ
tổ t tự ổ ố tữủ ự trồ t ừ
số t ỏ ổ ử ỹ tr ỹ
ừ t ồ t r t tự tr tự ữủ
t ồ st ữ t r rst r
r t tự ụ õ t ự
ữủ ữỡ ừ ồ ữ ữỡ
tỡ ữỡ tồ ở ữỡ số ự
t tự ự ợ ợ tự tờ qt t ữớ
t ổ ử ừ t t ỗ ó st ú
ự ỗ ữù ỗ ữù ồ s ọ
ử ừ t t tự ỹ tr
s tự số tổ ồ t t
tự ỹ tr s tự số
ởt số t tự ỹ tr s
tự số ũ ởt số q
ỗ t
3
ữỡ
ữỡ tự tự q
ữỡ t tự s tự số
ữỡ t ỹ tr s tự số
ử ừ t st ởt số ợ t tự
ỹ tr s tự số t rở ừ ú
ử tr st t ỹ tr q
tọ ỏ t ỡ s s tợ
t t ữợ ú ù t tr sốt q tr ồ t
ự ụ tọ ỏ t ỡ t
tợ ổ tr trữớ ồ ồ ồ
ú ù t tr sốt tớ ồ
t t rữớ
ỗ tớ t ụ ỷ ớ ỡ tợ ỗ
ổ ổ ú ù ở tổ tr tớ ồ t tr q
tr t
t
ữỡ ổ ứ
✸
❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝
❧✐➯♥ q✉❛♥
▼ö❝ ✤➼❝❤ ❝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 ✤÷ñ❝ ❞➔♥❤ ✤➸ ♥➯✉ ✈➲ ✤❛ t❤ù❝
❜➟❝ ❜❛ ✈➔ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ✤÷ñ❝
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
✤÷ñ❝ ❣å✐ ❧➔ ❤➺ sè✱
an
❧➔ ❤➺ sè ❝❛♦ ♥❤➜t ✈➔
a0
✭✶✳✶✮
❧➔ ❤➺ sè
tü ❞♦ ❝õ❛ ✤❛ t❤ù❝✳
fn (x) ❧➔ sè ♠ô ❝❛♦ ♥❤➜t ❝õ❛ ❧ô② t❤ø❛ ❝â ♠➦t tr♦♥❣ ✭✶✳✶✮
✈➔ ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔ 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❛ ❦❤æ♥❣ ✤à♥❤ ♥❣❤➽❛ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝
❦❤æ♥❣✮✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ✈î✐ ❤➺ sè ❧➜② tr♦♥❣ ✈➔♥❤
❤✐➺✉ ❧➔
A[x]✳
A=K
A
✤÷ñ❝ ❦þ
K[x] ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝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 ❝ì ❜↔♥
✣à♥❤ ❧þ ✶✳✶✳ ●✐↔ sû 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)
✈î✐
❝❤✐❛ ❤➳t ❝❤♦
a
❧➔ ♣❤➛♥ tû tò② þ ❝õ❛ ✈➔♥❤
n
þ ❝õ❛ ✈➔♥❤
A[x]✱
♣❤➛♥ tû
f (a) =
✈➔
r(x)
deg r(x) < deg g(x).
g(x)✳
n
●✐↔ sû
q(x)
A✱ f (x) =
ai x i
❧➔ ✤❛ t❤ù❝ tò②
i=0
ai ai
❝â ✤÷ñ❝ ❜➡♥❣ ❝→❝❤ 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✳
✤÷ñ❝ ❣å✐ ❧➔ ❣✐→ trà ❝õ❛
an xn + an−1 xn−1 + · · · + a1 x + a0 = 0 (an = 0).
✣à♥❤ ❧þ ✶✳✷✳ ●✐↔ sû A ❧➔ ♠ët tr÷í♥❣✱ a ∈ A ✈➔ f (x) ∈ A[x]✳ ❉÷ sè ❝õ❛
♣❤➨♣ ❝❤✐❛
f (x)
❝❤♦
x−a
❝❤➼♥❤ ❧➔
f (a)✳
✣à♥❤ ❧þ ✶✳✸✳ 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 ✳
●✐↔ sû
A
❧➔ ♠ët tr÷í♥❣✱
❧î♥
❦❤✐
r trữớ ủ
m = 1 t t ồ a ỡ ỏ m = 2 t a
ữủ ồ ố ừ ởt tự tờ số
ừ tự õ ở ừ õ ữớ t ởt
tự õ ởt ở
m
ữ ởt tự õ
m
trũ
ữủ ỗ rr
sỷ
f (x) = an xn + an1 xn1 + ã ã ã + a1 x + a0 A[x]
ợ
A
ởt trữớ õ tữỡ ú ừ
ởt tự õ
n 1
f (x)
(x a)
õ
q(x) = bn1 xn1 + ã ã ã + b1 x + b0 ,
tr õ
bn1 = an , bk = abk+1 + ak+1 , k = 0, . . . , n 2,
ữ số
r = ab0 + a0 .
ỵ
t
sỷ ữỡ tr
an xn + an1 xn1 + ã ã ã + a1 x + a0 = 0 (an = 0)
õ
n
tỹ ự
x1 , x2 , . . . , xn
t
E1 (x) := x1 + x2 + ã ã ã + xn
E2 (x) := x1 x2 + x1 x3 + ã ã ã + xn1 xn
.........
En (x) := x1 x2 . . . xn
ữủ số
x1 , x2 , . . . , xn
an1
=
an
an2
=
an
......
a0
= (1)n .
an
tọ tr t ú
ừ ữỡ tr õ
k
t tự k õ Cn số
n
t tr ừ
E1 (x), E2 (x), . . . , En (x) ữủ ồ tự ố ự
1, 2, . . . , n tữỡ ự
sỡ t
ỵ ộ tự tỹ n õ ổ q n tỹ
q tự õ ổ số tự ổ
q tự õ n ũ ởt tr ữ
t
n+1
t ừ ố số t õ tự
q tự n n + 1 trũ t n + 1
t ừ ố số t ú ỗ t
ỵ ồ tự f (x) R[x] õ n õ số số
an = 0
t
õ t t t t tỷ
m
s
i=1
ợ
(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
ồ
x0
ừ tự tọ t
tự
|x0 | 1 +
am
A
,
|a0 |
A = max |ak |.
1kn
số t ừ tự t số
n
1+
tr ừ ữỡ ừ tự tr õ
B
B
am
tr
ợ t ừ ổ số
fn (x) t ữợ fn (x) = g(x)q(x) ợ
deg(g) > 0 deg(q) > 0 t t õ g ữợ ừ fn (x) t t g(x)|fn (x)
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 tự 0 t
t õ r ú tố ũ t (f (x), h(x)) = 1
tự
ỵ ừ tự f (x) h(x) tố
ũ tỗ t tự
u(x)
v(x)
s
f (x)u(x) + h(x)v(x) 1.
t
tự
g(x)h(x)
tự
f (x)
h(x)
g(x)
tố ũ
tố ũ t tự
f (x)
ụ tố ũ
t
f (x)h(x)
f (x)
t
t
g(x)
f (x), g(x), h(x) tọ
g(x) g(x) h(x) tố ũ t f (x)
tự
✼
❚➼♥❤ ❝❤➜t ✶✳✸✳
◆➳✉ ✤❛ t❤ù❝
✈î✐
♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ 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➩ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈î✐ ♠å✐
m, n
♥❣✉②➯♥ ❞÷ì♥❣✳
▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ✤↕✐ sè ❝ì ❜↔♥
❚r♦♥❣ ♣❤➛♥ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ✤❛ t❤ù❝
✤↕✐ sè ❝ì ❜↔♥✳
✣à♥❤ ❧þ ✶✳✽
●✐↔ sû
✳
✭❇➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ 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 ✈➔ ✤÷ñ❝ ❣å✐
❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❈æs✐ ✭❈❛✉❝❤②✮✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ♥÷î❝ ♥❣♦➔✐
❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝â t➯♥ t✐➳♥❣ ❆♥❤ ❧➔ ✏❆▼✲●▼ ■♥❡q✉❛❧✐t②✑✱ ❝❤♦ ♥➯♥ ✈➲
s❛✉✱ t❛ ❣å✐ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❧➔ ✑❇➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉❣ ❜➻♥❤ ❝ë♥❣ ✈➔
tr✉♥❣ ❜➻♥❤ ♥❤➙♥✑✳
❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❦❤→ q✉❡♥ t❤✉ë❝ ✈î✐ ✤❛ sè ❜↕♥ ✤å❝ ✈➔ ✤➣ ✤÷ñ❝ ❝❤ù♥❣
♠✐♥❤ tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉ ❜➡♥❣ t✐➳♥❣ ❱✐➺t✱ ♥➯♥ ❝❤ó♥❣ tæ✐ s➩ ❦❤æ♥❣ 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♥❣ ✤÷ì♥❣ ✈î✐
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❝ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
6
x3 y 2 z.
ỵ
t tự r
ợ số tỹ tũ ỵ
x1 , x2 , . . . , xn
y1 , y2 , . . . , yn
t ổ õ t
tự
(x21 + x22 + ã ã ã + x2n )(y12 + y22 + ã ã ã + yn2 ) (x1 y1 + x2 y2 + ã ã ã + xn yn )2 .
(x1 , x2 , . . . , xn )
i = 1, n.
tự r
ở t tự
ỵ
xi = kyi
t tự r
(y1 , y2 , . . . , yn )
x1 , x2 , . . . , xn y1 , y2 , . . . , yn
i = 1, 2, . . . , n ổ õ t tự
số tỹ tr õ
yi > 0,
(x1 + x2 + ã ã ã + xn )2
x2 x2
x2
1 + 2 + ã ã ã + n.
y1 + y2 + ã ã ã + 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
õ
t õ s
y1 y 2 . . . y n
t
1
(x1 + x2 + . . . + xn )(y1 + y2 + . . . + yn ).
n
x 1 y1 + x 2 y 2 + ã ã ã + x n yn
y 1 y2 . . . yn
t
x 1 y1 + x 2 y 2 + ã ã ã + x n yn
tự r
1
(x1 + x2 + . . . + xn )(y1 + y2 + . . . + yn ).
n
x1 = x2 = . . . = xn
y1 = y2 = . . . = yn
ỵ
t tự s
sỷ số
tử tr
I(a, b) tr õ I(a, b) ữủ
ởt tr số t [a, b], [a, b), (a, b], (a, b) õ
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).
tự ởt số tự ỡ
ử tr ởt số tự ỡ ừ tự
ổ tự t ữỡ tr
ũ ữỡ tr tờ qt ổ ữủ ợ t
ờ tổ ữ t q ữỡ tr
tữớ tr t ồ t ồ s ọ r ử
tr ởt số t q ổ tự t ừ tự
ỵ
x1 , x2 , x3
ổ tự t
ừ ữỡ tr
ax3 + bx2 + cx + d = 0 (a = 0),
t
1 := x1 + x2 + x3
2 := x1 x2 + x1 x3 + x2 x3
3 := 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 số ụ tứ ũ ừ
x
ừ tự tr
s r ự
ỵ t ữỡ tr
x3 + ax2 + bx + c = 0
ợ số số tỹ
= 4a3 c + a2 b2 + 18abc 4b3 27c2
ữủ ồ t tự ừ ữỡ tr õ
> 0
t tt
x1 , x2 , x3
số tỹ
< 0 t ởt ừ ữỡ tr tỹ ỏ
ự ủ ũ
= 0 a2 3b = 0 t ữỡ tr õ tỹ
tr õ õ trũ ỏ
tr
=0
a2 3b = 0
t ữỡ tr õ
tỹ ũ ở
ự sỷ
x1 , x2 , x3
ừ ữỡ tr õ
t số ự ữ t t õ ởt tỹ õ t
ổ tự t ữỡ tr t õ
1 = x1 + x2 + x3 = a, 2 = x1 x2 + x1 x3 + x2 x3 = b, 3 = x1 x2 x3 = c.
t ữỡ ừ tự ố ự ỡ t ừ
x1 , x2 , x3
= T 2 = (x1 x2 )2 (x1 x3 )2 (x2 x3 )2 .
ứ tự t tr t õ
= 4a3 c + a2 b2 + 18abc 4b3 27c2 .
ó r tt tỹ t
tỹ ổ õ
2
= T > 0
T
số
ữủ ữủ s r tứ
ữợ
sỷ
x1
tỹ ỏ
x2 = + i
x3 = i
x2 , x3
ự ủ õ
õ t õ
T = (x1 i)(x1 + i)2i = 2i[(x1 )2 + 2 ].
õ
= T 2 = 4 2 [(x1 )2 + 2 ] < 0.
ứ t q tr tr t t ữỡ tr
õ t ừ ữỡ tr tỹ
= 0
s tọ õ
ở t t tự
1
= (x1 x2 )2 + (x2 x3 )2 + (x3 x1 )2 = 2(12 32 ) = 2(a2 3b).
ó r
x1 , x2 , x3
số tỹ t
1
= 0
tự
a2 = 3b
ữỡ tr õ tỹ ở
= 0 a2 = 3b t ữỡ tr õ ở ỏ
= 0 a2 = 3b t ữỡ tr õ số ỵ ữủ ự
ử
ởt ữỡ tr õ
ữỡ ừ ữỡ 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 ✳ ❚❤❡♦ ✣à♥❤
❧þ ❱✐➧t❡ t❛ ❝â
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.
❚❤❡♦ ✣à♥❤ ❧þ ❱✐➧t❡✱ t❛ ❝â
b
b
σ1 = 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.
❚❤❡♦ ✣à♥❤ ❧þ ❱✐➧t❡✱ t❛ ❝â
σ1 = 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♣ ❞ö♥❣ ❝æ♥❣ 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➻♥❤ ✭✶✳✾✮✳ ❚❤❡♦ ❝æ♥❣ 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✳
❱î✐
⇔
σ13 − 3σ1 σ2 + 3σ3 = −a3 ,
σ23 − 3σ1 σ2 σ3 + 3σ32 = b3 ,
σ = −c3 .
3
σ1 , σ2 , σ3 tø ✭✶✳✶✵✮ ✈➔♦ ❤➺ tr➯♥ ✈➔ rót ❣å♥✱ t❛ t❤✉ ✤÷ñ❝
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➻♠ ✤÷ñ❝ ❝→❝ ❣✐→ trà ❝õ❛ σ1 , σ2 , σ3 ✱
trà ❝õ❛ ❝→❝ ➞♥ sè x, y, z ✳ ✣✐➲✉ ♥➔② ❞➵ ❞➔♥❣ t❤ü❝ ❤✐➺♥
❞➔♥❣ t➻♠ ✤÷ñ❝ ♥❣❤✐➺♠
❝➛♥ ♣❤↔✐ t➻♠ ❝→❝ ❣✐→
✤÷ñ❝ ♥❤í ✤à♥❤ ❧➼ s❛✉ ✤➙②✳
ỵ
sỷ 1, 2, 3 số tỹ õ õ
ữỡ tr
u3 1 u2 + 2 u 3 = 0
x+y+z
= 1 ,
xy + yz + zx = 2 ,
xyz
= 3 .
ữỡ tr
ợ ữ s
u1 , u2 , u3
ừ ữỡ tr
t õ
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 ổ ỏ ữủ
z = c
ừ t số
a, b, c
x = a, y = b,
ừ ữỡ
tr
ự
sỷ
u1 , u2 , u3
ừ ữỡ tr
õ t õ ỗ t tự
u3 1 u2 + 2 u 3 = (u u1 )(u u2 )(u u3 ).
ứ õ t õ tự t
u + u2 + u3
1
u1 u2 + u1 u3 + u2 u3
u u u
1 2 3
r
u1 , u2 , u3
= 1 ,
= 2 ,
= 3 .
ừ r ỏ
ỳ ữủ tr ừ số
✶✺
✭✶✳✶✹✮ ❦❤æ♥❣ ❝á♥ ♥❣❤✐➺♠ ♥➔♦ ❦❤→❝ s➩ ✤÷ñ❝ ❧➔♠ s→♥❣ tä ❞÷î✐ ✤➙②✳
●✐↔ sû
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
❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛
✭✶✳✶✸✮✳ ✣à♥❤ ❧þ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
✣à♥❤ ❧þ ✶✳✶✻✳ ●✐↔ sû σ1, σ2, σ3 ❧➔ ❝→❝ 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❤➻
σ1 ≥ 0,
❈❤ù♥❣ ♠✐♥❤✳
❧þ ✶✳✶✺✱
x, y, z
●✐↔ sû
✭✶✳✶✺✮
x, y, z
σ2 ≥ 0,
σ3 ≥ 0.
❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✹✮✳ ❑❤✐ ✤â t❤❡♦ ✣à♥❤
❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮✳ ❚❤❡♦ ✣à♥❤ ❧þ ✶✳✶✹✱
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮ ❝â ♥❣❤✐➺♠ t❤ü❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❜✐➺t t❤ù❝ ❝õ❛ ♥â ❦❤æ♥❣
x, y, z ❧➔ ❦❤æ♥❣ ➙♠✱
t❤➻ ❤✐➸♥ ♥❤✐➯♥ σi ≥ 0
(i = 1, 2, 3)✳ ◆❣÷ñ❝ ❧↕✐✱ ♥➳✉ σi ≥ 0 (i = 1, 2, 3) ✈➔
➙♠✱ ♥❣❤➽❛ ❧➔ ✭✶✳✶✺✮ ✤÷ñ❝ t❤ä❛ ♠➣♥✳ ◆❣♦➔✐ r❛✱ ♥➳✉ ❝→❝ sè
✭✶✳✶✺✮ ✤÷ñ❝ 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➻♥❤ ✭✶✳✶✻✮ ❦❤æ♥❣ t❤➸ ❝â ♥❣❤✐➺♠
❞÷ì♥❣✱ ❞♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮ ❦❤æ♥❣ t❤➸ ❝â ♥❣❤✐➺♠ ➙♠✳ ❚ø ✤â s✉② r❛
x, y, z
❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ✣à♥❤ ❧þ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❱➼ ❞ö ✶✳✻✳
●✐↔✐ ❤➺ ♣❤÷ì♥❣ 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 ❞ö♥❣
❝æ♥❣ 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➻♠ ✤÷ñ❝
t❛ ❝â
x, y, z
σ1 = 2, σ2 = −1, σ3 = −2.
❚❤❡♦ ✣à♥❤ ❧þ ✶✳✶✺✱
❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ 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❤➔♥❤ ♥❤➙♥ tû✱ tr÷î❝ ❤➳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 ♥❤➟♥ ✤÷ñ❝ tø h(x, y, z) ❜➡♥❣ ❝→❝❤ ❤♦→♥ ✈à ❝→❝ ❜✐➳♥ x, y, z ♥❣❤➽❛
❝ì sð
σ1 , σ2 , σ3
✤➸ ✤÷ñ❝ ✤❛ 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❤➼ ❞ö ✤è✐ ✈î✐
x, y
g(x, y, z)
❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝❤➾ ✈î✐
♥❣❤➽❛ ❧➔
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) ❦❤æ♥❣ ❝â t➼♥❤ ✤è✐ ①ù♥❣✳
✣è✐ ✈î✐ ✤❛ 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✐ ✈î✐ ✤❛ 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✐ ✈î✐ 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→❝
✤÷ñ❝ 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♥❣✳
❱➼ ❞ö ✶✳✾✳
❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈î✐ ♠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✱ ♥➯♥ ♥â ❝â
▲í✐ ❣✐↔✐✳ ❚r÷î❝ ❤➳t t❛ ♣❤➙♥ t➼❝❤
❞↕♥❣
g(x, y, z) = a(x + y)(x + z)(y + z).
❈❤♦
x=y=z=1
t❛ t➻♠ ✤÷ñ❝
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).
❇➡♥❣ ❝→❝❤ t÷ì♥❣ tü t❛ t❤➜②
✈î✐ ♠å✐
n
♥❣✉②➯♥ ❞÷ì♥❣✱
❱➼ ❞ö ✶✳✶✵✳
❝❤♦
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✳ ❚❛ ❝â
❚❤❡♦ ✣à♥❤ ❧þ ❇❡③♦✉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
❱î✐ p =
− b, q = −
+
− c.
3
27
3
✶✮ ◆➳✉ p = 0 ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳
p
✷✮ ◆➳✉ p > 0✳ ✣➦t y = 2
t✳ ❑❤✐ ✤â✱ t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤
3
√
3
3q
4t3 − 3t = m ✈î✐ m = √ .
2p p
y−
❛✮ ◆➳✉
❜✮ ◆➳✉
|m| ≤ 1✱
a
3
P❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â t❤➸ ✈✐➳t ❞÷î✐
3
✤➦t
+a y−
m = cos α✱
♣❤÷ì♥❣ 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❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤
4t3 + 3t = m
✣➦t
3
✈î✐
✱ tr♦♥❣ ✤â
√
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 ❝→❝ ✤à♥❤ ❧þ ❝ì ❜↔♥ tr♦♥❣ t❛♠ ❣✐→❝ s❛✉✳
✣à♥❤ ❧þ ✶✳✶✼
✭✣à♥❤ ❧þ ❤➔♠ sè
❚r♦♥❣ t❛♠ ❣✐→❝
ABC
✳
sin✮
t❛ ❧✉æ♥ ❝â
a
b
c
=
=
= 2R.
sin A sin B
sin C
√
m2 + 1 .
ỵ
ỵ số s
r t t ổ õ
a2 = b2 + c2 2bc cos A;
b2 = a2 + c2 2ac cos B;
c2 = a2 + b2 2ab cos C.
t
ABC
ở ừ t
sỷ ữủt
a, b, c
ừ ữỡ tr
t3 2pt2 + (p2 + r2 + 4Rr)t 4pRr = 0.
a, b, c ở ừ t
ự sỷ
t ỵ số
sin
ổ tự õ ổ t õ
a = 2R sin A = 4R sin
A
A
cos .
2
2
ổ tự t ữớ trỏ ở t t
A
A
2.
p a = r cot = r
A
2
sin
2
cos
t ữủ
sin2
A
ar
=
.
2
4R(p a)
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 số t ữủ
r2 a + a(p a)2 = 4Rr(p a)
a3 2pa2 + (p2 + r2 + 4Rr)a 4pRr = 0.
a
ừ ữỡ tỹ
õ ự
b, c
ụ ừ
✷✷
❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤
❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t❛ q✉❛♥ t➙♠ ❝❤õ ②➳✉ ✤➳♥ ❝→❝ ❞↕♥❣ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
✈➔ ✤❛ t❤ù❝ ✤ç♥❣ ❜➟❝ ❜✐➳♥ sè t❤ü❝ ✈➔ ♥❤➟♥ ❣✐→ trà t❤ü❝✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤
❝õ❛ ❝❤÷ì♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ tø t➔✐ ❧✐➺✉ ❬✸❪✳
✷✳✶ ❇➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ✤❛ t❤ù❝ ❜➟❝ ❜❛
✷✳✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳
▼ët ✤ì♥ t❤ù❝
ϕ(x, y, z)
❝õ❛ ❝→❝ ❜✐➳♥
x, y, z
✤÷ñ❝ ❤✐➸✉
❧➔ ❤➔♠ sè ❝â ❞↕♥❣
ϕ(x, y, z) = aklm xk y l z m ,
k, l, m ∈ N ✤÷ñ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ❝→❝ ❜✐➳♥ x, y, z ❀ sè aklm ∈ R∗ ✤÷ñ❝
❣å✐ ❧➔ ❤➺ sè ❝õ❛ ✤ì♥ t❤ù❝✱ ❝á♥ sè k + j + m ✤÷ñ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ✤ì♥ t❤ù❝
ϕ(x, y, z)✳
tr♦♥❣ ✤â
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳
▼ët ❤➔♠ sè
P (x, y, z)
❝õ❛ ❝→❝ ❜✐➳♥
x, y, z
✤÷ñ❝ ❣å✐ ❧➔
♠ët ✤❛ t❤ù❝✱ ♥➳✉ ♥â ❝â t❤➸ ✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ð ❞↕♥❣ tê♥❣ ❤ú✉ ❤↕♥ ❝→❝ ✤ì♥
t❤ù❝✿
aklm xk y l z m
P (x, y, z) =
k+l+m≤n
❇➟❝ ❧î♥ ♥❤➜t ❝õ❛ ❝→❝ ✤ì♥ t❤ù❝ tr♦♥❣ ✤❛ t❤ù❝ ✤÷ñ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝✳
✣à♥❤ ♥❣❤➽❛ ✷✳✸✳
P (x, y, z) ✤÷ñ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ♥➳✉ ♥â
❤♦→♥ ✈à ❝õ❛ x, y, z ♥❣❤➽❛ ❧➔
✣❛ t❤ù❝
❦❤æ♥❣ t❤❛② ✤ê✐ ✈î✐ ♠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✳