Đồng nhất thức newton girard và ứng dụng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❇Ị■ ❚❍➚ ❍❷■ ❨➌◆ ✣➬◆● ◆❍❻❚ ❚❍Ù❈ ◆❊❲❚❖◆ ✲ ●■❘❆❘❉ ❱⑨ Ù◆● ❉Ö◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❇Ò■ ❚❍➚ ❍❷■ ❨➌◆ ✣➬◆● ◆❍❻❚ ❚❍Ù❈ ◆❊❲❚❖◆ ✲ ●■❘❆❘❉ ❱⑨ Ù◆● ❉Ư◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ ❚♦→♥ ❝➜♣ ▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❙✳ ❚❘❺◆ ◆●❯❨➊◆ ❆◆ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✸ ✶✳✶ ✣❛ t❤ù❝ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❈❤✉é✐ ❧ô② t❤ø❛ ❤➻♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ tự trữ ỵ t ỗ t tự trr ự ỵ ỡ ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ ỗ t tự trr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ỗ t tự trr tờ ụ tứ ❝õ❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ỗ t tự trr ỵ số ụ ỗ t tự trr ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺✳✶ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺✳✷ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤➔♥❤ ♥❤➙♥ tû ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✺✳✸ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ✳ ✳ ✹✷ ✷✳✺✳✹ ❚➻♠ ♥❣❤✐➺♠ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✺✳✺ ❈❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✺✳✻ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✺✳✼ ❚rö❝ ❝➠♥ t❤ù❝ ð ♠➝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ❑➌❚ ▲❯❾◆ ❚➔✐ ❧✐➺✉ t é ỗ t tự ◆❡✇t♦♥✲●✐r❛r❞ ❝❤♦ t❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ tê♥❣ ❧ô② t❤ø❛ ❝→❝ ❜✐➳♥ ✈➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥✳ ỹ ỗ t tự t ữủ tê♥❣ ❧ô② t❤ø❛ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ P (x) q số õ ỗ t tự ữủ t r s t ỵ t÷ð♥❣ ♥➔② ❝ơ♥❣ ✤÷đ❝ ❝❤♦ ❧➔ ①✉➜t ❤✐➺♥ tr♦♥❣ ❝ỉ♥❣ tr trữợ õ rt r õ t tữớ ỗ t tự trr ỗ t tự trr ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ữ ỵ tt s ỵ tt t ỵ t❤✉②➳t tê ❤đ♣ ❝ơ♥❣ ♥❤÷ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ❝õ❛ ✤í✐ sè♥❣✳ ▲✉➟♥ ✈➠♥ ♥➔② t➻♠ ❤✐➸✉ ♠ët sè ❝→❝❤ ự ỗ t tự trr ự tr ❣✐↔✐ t♦→♥ ❝➜♣✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ✤❛ t❤ù❝✱ ❝❤✉é✐ ❧ô② t❤ø❛ ❤➻♥❤ t❤ù❝✱ ♠❛ tr tự trữ ỵ t ữỡ ữỡ tr ỗ t tự ◆❡✇t♦♥✲●✐r❛r❞ ✈➔ ♠ët sè ù♥❣ ❞ö♥❣✳ ✣➸ ❝â ❝→❝❤ ♥❤➻♥ tê♥❣ q✉❛♥ ✈➲ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ♠ö❝ ✤➛✉ ❝õ❛ ữỡ tr ỵ ỡ tự ✤è✐ ①ù♥❣ ✈➔ ♠ët sè t❤✉➟t t♦→♥ ❝ì ❜↔♥ ❜✐➸✉ ❞✐➵♥ ♠ët ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ q✉❛ ❝→❝ ✤❛ t❤ù❝ ố ự ỡ ỗ t tự trr ợ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ✈➔ ♥❤✐➲✉ ❞↕♥❣ ❦❤→❝ ♥❤❛✉ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♠ư❝ t❤÷ ❤❛✐ ❝õ❛ ❝❤÷ì♥❣ ♥➔②✳ ✣➦❝ ❜✐➺t ởt số ự ữ ự ỵ số ♥❣ô ❣✐→❝✱ t➼♥❤ ♠ët sè ❜✐➸✉ t❤ù❝ ✤è✐ ①ù♥❣✱ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû✱ ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝✱ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ trư❝ ❝➠♥ t❤ù❝ ð ♠➝✉✱ ✳✳✳ ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❧➔ ❝✉è♥ s→❝❤ ❬✷❪ ❝õ❛ ●❙✳ ▲➯ ❚❤❛♥❤ ◆❤➔♥ ✈➔ ❝→❝ ❜➔✐ ❜→♦ ❬✻❪✱ ❬✼❪✱ ❬✽❪ ✈➔ ♠ët sè t➔✐ ❧✐➺✉ æ♥ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ð ♣❤ê t❤æ♥❣✳ ❚r♦♥❣ q✉→ tr➻♥❤ tổ ữủ sỹ ữợ ❣✐ó♣ ✤ï t➟♥ t➻♥❤ ❝õ❛ ❚❙✳ ❚r➛♥ ◆❣✉②➯♥ ❆♥✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛②✳ ✶ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ qỵ t ổ ợ õ ✾ ✤➣ tr✉②➲♥ t❤ư ✤➳♥ ❝❤♦ tỉ✐ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✼ ❇ò✐ ❚❤à ❍↔✐ ❨➳♥ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ s✉èt ❝❤÷ì♥❣ ♥➔②✱ ❧✉ỉ♥ ❣✐↔ t❤✐➳t N0 = {0, 1, 2, } ✤ì♥ ✈à✳ ❚❛ ❦➼ ❤✐➺✉ N = {1, 2, 3, } ❜ë n V ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ❧➔ t➟♣ ❝→❝ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ ✈➔ ❧➔ t➟♣ ❝→❝ sè tü ♥❤✐➯♥✳ ❱ỵ✐ n ∈ N✱ t❛ ❦➼ ❤✐➺✉ Nn0 ❧➔ t➟♣ ❝→❝ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ ✶✳✶ ✣❛ t❤ù❝ ♥❤✐➲✉ ❜✐➳♥ ▼é✐ ❜ë n sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠ ✤ì♥ t❤ù❝ xi11 · · · xinn ❝õ❛ n ❜✐➳♥ x1 , , xn tữớ t ỡ tự ữợ ❞↕♥❣ ❱ỵ✐ i = j, j = (j1 , , jn ) ∈ Nn0 , tù❝ ❧➔ ik = jk ✈ỵ✐ ♠å✐ ❝õ❛ tø✮ ✈➔ xi ❧➔ ♠ët ✤ì♥ t❤ù❝ ✤÷đ❝ ❣å✐ ❧➔ ▼ët ❧➔ i1 + · · · + in ❈❤ó♥❣ t❛ xi ✈➔ xj ❧➔ ❜➡♥❣ ♥❤❛✉ ♥➳✉ k✳ ❧➔ ♠ët ❜✐➸✉ t❤ù❝ õ ỡ tự tứ ỗ ❞↕♥❣ ✈ỵ✐ ❝❤♦ t❛ ♠ët xi ▼ët ❧➔ i = (i1 , · · · , in ) ∈ Nn0 axi ợ aV ữủ ỡ tự tø✳ ❤➺ sè ❍❛✐ tø ✤÷đ❝ ❣å✐ ♥➳✉ ❤❛✐ ✤ì♥ t❤ù❝ ❝õ❛ ❝❤ó♥❣ ❜➡♥❣ ♥❤❛✉✳ ❍❛✐ tø ✤÷đ❝ ❣å✐ ❧➔ ú ỗ õ ũ số t❤ù❝ ❧➔ ♠ët tê♥❣ ❝õ❛ ❤ú✉ ❤↕♥ tø✳ ◆➳✉ u = axi v = bxi tứ ỗ t t õ t ữợ ữủ tờ ú u + v = (a + b)xi ❱➻ ✈➟②✱ ữợ ữủ tứ ỗ ộ t❤ù❝ ♠ët ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ t➢❝ x i f (x1 , , xn ) = i∈Nn0 ✸ f (x1 , , xn ) õ t tờ tứ ổ ởt ổ ỗ ❞↕♥❣✱ tr♦♥❣ ✤â ❝❤➾ ❝â ❤ú✉ ❤↕♥ tø ❦❤→❝ ✭tù❝ ❧➔ ❤➺ sè ❝õ❛ tø ❦❤→❝ 0✮✱ ✈➔ ❜✐➸✉ ❞✐➵♥ ♥➔② ❧➔ ❞✉② ♥❤➜t ♥➳✉ ❦❤æ♥❣ ❦➸ ✤➳♥ t❤ù tü ❝→❝ ❤↕♥❣ tû✳ ▼é✐ tø ❦❤→❝ ❝❤➼♥❤ t➢❝ ❝õ❛ ✤❛ t❤ù❝ ✤÷đ❝ ❣å✐ ❧➔ x i ❍❛✐ ✤❛ t❤ù❝ ①✉➜t ❤✐➺♥ tr♦♥❣ ❜✐➸✉ ❞✐➵♥ ♠ët tø ❝õ❛ ✤❛ t❤ù❝ ✤â✳ bi xi ❜➡♥❣ ♥❤❛✉ ✈➔ i∈Nn0 ❧➔ i∈Nn0 ♥➳✉ = bi ✈ỵ✐ ♠å✐ i ∈ Nn0 ✳ ❇➟❝ ❝õ❛ ♠ët tø ❦❤→❝ ❧➔ ❜➟❝ ❝õ❛ ✤ì♥ t❤ù❝ ❝õ❛ tø ✤â✳ ❇➟❝ ✭❤❛② ❜➟❝ tê♥❣ t❤➸✮ ❝õ❛ ✤❛ t❤ù❝ f (x1 , , xn ) = 0, ❦➼ ❤✐➺✉ ❜ð✐ deg f (x1 , , xn ), ❧➔ sè ❧ỵ♥ ♥❤➜t tr♦♥❣ ❝→❝ ❜➟❝ ❝õ❛ ❝→❝ tø ❝õ❛ ❜➟❝ ❝❤♦ ✤❛ t❤ù❝ t❤ù❝ ❜➟❝ 0✳ ✣❛ t❤ù❝ ❤➡♥❣ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❞↕♥❣ ❜➟❝ m✮ ❧➔ ✤❛ t❤ù❝ ❚❛ ❦❤æ♥❣ ✤à♥❤ ♥❣❤➽❛ ❤♦➦❝ ✤❛ t❤ù❝ ❜➟❝ 0✳ ❈→❝ ✤❛ ✤❛ t❤ù❝ t✉②➳♥ t➼♥❤✳ ✣❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ m ❧➔ ♠ët ✤❛ t❤ù❝ ♠➔ ❝→❝ tø ❝õ❛ ♥â ✤➲✉ ❝â ❜➟❝ t❤✉➛♥ ♥❤➜t ❜➟❝ ❤❛✐ ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ t❤❡♦ ❜✐➳♥ xk f (x1 , , xn ) t ữỡ ợ ộ m ✣❛ t❤ù❝ k ∈ {1, , n}, ❝õ❛ ♠ët ✤❛ t❤ù❝ ❧➔ sè ❧ỵ♥ ♥❤➜t tr♦♥❣ ❝→❝ sè ♠ô ❝õ❛ xk ①✉➜t ❤✐➺♥ tr♦♥❣ ❝→❝ tø ❝õ❛ ✤❛ t❤ù❝ ✤â✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❑➼ ❤✐➺✉ V [x1, , xn] ❧➔ t➟♣ ❝→❝ ✤❛ t❤ù❝ n ❜✐➳♥ x1, , xn ✈ỵ✐ ❤➺ sè tr♦♥❣ V ✳ ❱ỵ✐ i, j ∈ Nn0 , tr♦♥❣ ✤â i = (i1 , , in ) ✈➔ j = (j1 , , jn ), t❛ ✤à♥❤ ♥❣❤➽❛ i + j = (i1 + j1 , , in + jn ) ❑❤✐ ✤â V [x1 , , xn ] ❧➔ ♠ët ✈➔♥❤ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ x i i∈Nn0 ✈ỵ✐ ♠å✐ ✤❛ t❤ù❝ bi xi = i∈Nn x i , i∈Nn0 i∈Nn0 i∈Nn0 i∈Nn0 (ai + bi )xi ; bi xi = x i + i∈Nn0 ck xk , ck = k∈Nn0 b j i+j=k bi xi ∈ V [x1 , , xn ]✳ ❱➔♥❤ V [x1 , , xn ] ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ n ❜✐➳♥ x1 , , xn ✈ỵ✐ ❤➺ sè tr♦♥❣ V ✳ ◆❤➟♥ ①➨t ✶✳✶✳✷✳ ❇➡♥❣ q✉② ♥↕♣✱ ✈➔♥❤ ✤❛ t❤ù❝ n ❜✐➳♥ V [x1, , xn] ✈ỵ✐ ❤➺ sè tr♦♥❣ V ❝❤➼♥❤ ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ xn ✈ỵ✐ ❤➺ sè tr♦♥❣ ✈➔♥❤ V [x1 , , xn−1 ] ❚ø ✤à♥❤ ♥❣❤➽❛✱ t❛ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙② ✈➲ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝✳ ✹ ❇ê ✤➲ ✶✳✶✳✸✳ ❈❤♦ f1(x1, , xn), f2(x1, , xn) ∈ V [x1, , xn] ❧➔ ❝→❝ ✤❛ t❤ù❝ ❦❤→❝ s❛♦ ❝❤♦ tê♥❣ ✈➔ t➼❝❤ ❝õ❛ ❝❤ó♥❣ ✤➲✉ ❦❤→❝ 0✳ ❑❤✐ ✤â max{deg fi (x1 , , xn )} (i) deg(f1 (x1 , , xn ) + f2 (x1 , , xn )) (ii) deg f1 (x1 , , xn )f2 (x1 , , xn ) i=1,2 deg f1 (x1 , , xn ) + deg f2 (x1 , , xn ), ✈➔ ✤➥♥❣ tự r V ỵ ✶✳✶✳✹ ✳ ❈❤♦ R ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ✈➔ ✤❛ tự ỵ t f (x) R[x], R✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ α ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ f (x) ❧➔ f (x) ❝❤✐❛ ❤➳t ❝❤♦ (x − α)✳ ❚ø ❦➳t q✉↔ tr➯♥ t❛ ❝â ỗ r tự x a sû ✤❛ t❤ù❝ tr♦♥❣ R ❧➔ ♠✐➲♥ ♥❣✉②➯♥ R[x]✳ ❈❤✐❛ f (x) t❛ ❝â f (x) = an xn + · · · + a1 x + a0 ❝❤♦ g(x) = bn−1 xn−1 + · · · + b1 x + b0 ✱ f (x) ❞÷ x − a, a ∈ R✱ r ∈ R✳ ❱➻ ❧➔ ♠ët t❛ ✤÷đ❝ t❤÷ì♥❣ ❞↕♥❣ f (x) = (x − a)g(x) + r   bn−1 = an      ···    b = a + ab i−1 i i  ···      b0 = a1 + ab1     r = a0 + b ỡ ỗ ú t t t❤÷ì♥❣ tr♦♥❣ ✤â g(x) bi , i = 0, · · · , n − ✈➔ ❞÷ r ❝❤♦ ♥➯♥ ✭✶✳✶✮ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ f (x) ❝❤♦ x − a✱ ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ✶✳✶ ✤÷đ❝ ❣å✐ ❧➔ ỗ r an an1 bn1 bn2 ✶✳✶✳✺ ❚❛ ❣å✐ α ❧➔ ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ❝õ❛ (x − α)k+1 ✳✳✳ a1 a0 b0 r ✳ ❈❤♦ f (x) ∈ R[x], α ∈ R, k ∈ Z, k ≥ 1✳ f (x) ♥➳✉ f (x) ✭◆❣❤✐➺♠ ❜ë✐✮ ♥❣❤✐➺♠ ❜ë✐ k ✳✳✳ ❝❤✐❛ ❤➳t ❝❤♦ (x − α)k ♥❤÷♥❣ ♥❣❤➽❛ ❧➔✿ f (x) = (x − α)k g(x), ∀x ∈ R, g(α) = ◆➳✉ k = 1✱ α ♥❣❤✐➺♠ ❦➨♣✳ ❧➔ t❛ ❣å✐ α ❧➔ ♥❣❤✐➺♠ ✤ì♥ ❤❛② ❝á♥ ❣å✐ ✺ ♥❣❤✐➺♠✱ ♥➳✉ k = 2✱ t❛ ❣å✐ ❇ê ✤➲ ✶✳✶✳✻✳ ❈❤♦ f (x) ∈ R[x] P❤➛♥ tû a ∈ R ❧➔ ♥❣❤✐➺♠ ❜ë✐ k ❝õ❛ f (x) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f (x) = (x − a)k g(x) ✈ỵ✐ g(x) ∈ R[x] ✈➔ g(a) = ỵ R ởt = f (x) ∈ R[x] ✈➔ a1 , a2 , , ar ∈ R ❧➔ ❝→❝ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ❝õ❛ f (x) ●✐↔ sû ❧➔ ♥❣❤✐➺♠ ❜ë✐ ki ❝õ❛ f (x) ✈ỵ✐ i = 1, 2, , r ❑❤✐ ✤â t❛ ❝â f (x) = (x − a1 )k1 (x − a2 )k2 (x − ar )kr g(x) tr♦♥❣ ✤â g(x) ∈ R[x] ✈➔ g(ai ) = ✈ỵ✐ ♠å✐ i = 1, , r ❍➺ q✉↔ ✶✳✶✳✽✳ ❈❤♦ R ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ✈➔ f (x) ∈ R[x] ❧➔ ♠ët ✤❛ t❤ù❝ ❦❤→❝ ❑❤✐ ✤â sè ♥❣❤✐➺♠ ❝õ❛ f (x)✱ ♠é✐ ♥❣❤✐➺♠ t➼♥❤ ✈ỵ✐ sè ❜ë✐ ❝õ❛ ♥â✱ ❦❤ỉ♥❣ ✈÷đt q✉→ ❜➟❝ ❝õ❛ ❝õ❛ f (x) ❍➺ q✉↔ ✶✳✶✳✾✳ ❈❤♦ R ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ✈➔ f (x), g(x) deg(f (x)) n ✈➔ deg(g(x)) ∈ R[x]✱ tr♦♥❣ ✤â n ◆➳✉ f (x) ✈➔ g(x) ❝â ❣✐→ trà ❜➡♥❣ ♥❤❛✉ t↕✐ n + ♣❤➛♥ tû ❦❤→❝ ♥❤❛✉ ❝õ❛ R t❤➻ f (x) = g(x) ✣à♥❤ ỵ sỷ ỵ t t f (x) = an xn + an−1 xn−1 + + a1 x + a0 , ❧➔ ✤❛ t❤ù❝ ❜➟❝ n ❝â n ♥❣❤✐➺♠ α1 , , αn ❑❤✐ ✤â  an−1  α + α + + α = −  n   an   an−2   αi αj =   an  i