Teore ma Viye ta formuli nazvani na chest Fransua Viyeta sho virazhayut koeficiyenti mnogochlena cherez jogo koreni Ci formuli zruchno vikoristovuvati dlya perevirki pravilnosti znahodzhennya koreniv ta dlya zadannya mnogochlena z viznachenimi vlastivostyami Zmist 1 Formuli 2 Dovedennya 3 Prikladi 4 Div takozh 5 DzherelaFormulired Yaksho x 1 x 2 x n displaystyle x 1 x 2 ldots x n nbsp koreni mnogochlena x n a 1 x n 1 a 2 x n 2 a n displaystyle x n a 1 x n 1 a 2 x n 2 a n nbsp kozhen korin prisutnij vidpovidno do jogo kratnosti to koeficiyenti a 1 a n displaystyle a 1 ldots a n nbsp ye elementarnimi simetrichnimi mnogochlenami vid koreniv a same a 1 x 1 x 2 x n a 2 x 1 x 2 x 1 x 3 x 1 x n x 2 x 3 x n 1 x n a 3 x 1 x 2 x 3 x 1 x 2 x 4 x n 2 x n 1 x n a n 1 1 n 1 x 1 x 2 x n 1 x 1 x 2 x n 2 x n x 2 x 3 x n a n 1 n x 1 x 2 x n displaystyle begin matrix a 1 amp amp x 1 x 2 ldots x n a 2 amp amp x 1 x 2 x 1 x 3 ldots x 1 x n x 2 x 3 ldots x n 1 x n a 3 amp amp x 1 x 2 x 3 x 1 x 2 x 4 ldots x n 2 x n 1 x n cdots amp amp cdots a n 1 amp amp 1 n 1 x 1 x 2 ldots x n 1 x 1 x 2 ldots x n 2 x n ldots x 2 x 3 x n a n amp amp 1 n x 1 x 2 ldots x n end matrix nbsp Inshimi slovami 1 k a k displaystyle 1 k a k nbsp dorivnyuye sumi vsih mozhlivih k displaystyle k nbsp dobutkiv iz koreniv Yaksho starshij koeficiyent mnogochlena a 0 1 displaystyle a 0 neq 1 nbsp to dlya zastosuvannya formuli Viyeta neobhidno rozdiliti vsi koeficiyenti na a 0 displaystyle a 0 nbsp Iz ostannoyi formuli Viyeta viplivaye sho yaksho koreni mnogochlena ye cilimi to voni ye dilnikami jogo vilnogo chlena yakij takozh ye cilim Dovedennyared Dovedennya vikoristovuye rivnist x n a 1 x n 1 a 2 x n 2 a n x x 1 x x 2 x x n displaystyle x n a 1 x n 1 a 2 x n 2 ldots a n x x 1 x x 2 ldots x x n nbsp Prava chastina predstavlyaye mnogochlen rozkladenij na mnozhniki Pislya rozkrittya duzhok koeficiyenti pri odnakovih stepenyah x povinni buti odnakovimi v oboh chastinah rivnosti z chogo sliduyut formuli Viyeta Prikladired Yaksho x 1 x 2 displaystyle x 1 x 2 nbsp koreni kvadratnogo rivnyannya a x 2 b x c 0 displaystyle ax 2 bx c 0 nbsp to x 1 x 2 b a x 1 x 2 c a displaystyle x 1 x 2 frac b a qquad x 1 x 2 frac c a nbsp V chastkovomu vipadku pri a 1 displaystyle a 1 nbsp kvadratne rivnyannya x 2 p x q 0 displaystyle x 2 px q 0 nbsp to x 1 x 2 p x 1 x 2 q displaystyle x 1 x 2 p qquad x 1 x 2 q nbsp Yaksho x 1 x 2 x 3 displaystyle x 1 x 2 x 3 nbsp koreni kubichnogo rivnyannya a x 3 b x 2 c x d 0 displaystyle ax 3 bx 2 cx d 0 nbsp to x 1 x 2 x 3 b a x 1 x 2 x 1 x 3 x 2 x 3 c a x 1 x 2 x 3 d a displaystyle x 1 x 2 x 3 frac b a qquad x 1 x 2 x 1 x 3 x 2 x 3 frac c a qquad x 1 x 2 x 3 frac d a nbsp V chastkovomu vipadku kubichne rivnyannya x 3 p x q 0 displaystyle x 3 px q 0 nbsp to x 1 x 2 x 3 0 x 1 x 2 x 1 x 3 x 2 x 3 p x 1 x 2 x 3 q displaystyle x 1 x 2 x 3 0 qquad x 1 x 2 x 1 x 3 x 2 x 3 p qquad x 1 x 2 x 3 q nbsp Yaksho x 1 x 2 x 3 x 4 displaystyle x 1 x 2 x 3 x 4 nbsp koreni rivnyannya chetvertogo stepenya a x 4 b x 3 c x 2 d x e 0 displaystyle ax 4 bx 3 cx 2 dx e 0 nbsp to x 1 x 2 x 3 x 4 b a x 1 x 2 x 1 x 3 x 1 x 4 x 2 x 3 x 2 x 4 x 3 x 4 c a displaystyle x 1 x 2 x 3 x 4 frac b a qquad x 1 x 2 x 1 x 3 x 1 x 4 x 2 x 3 x 2 x 4 x 3 x 4 frac c a nbsp x 1 x 2 x 3 x 1 x 2 x 4 x 1 x 3 x 4 x 2 x 3 x 4 d a x 1 x 2 x 3 x 4 e a displaystyle x 1 x 2 x 3 x 1 x 2 x 4 x 1 x 3 x 4 x 2 x 3 x 4 frac d a qquad x 1 x 2 x 3 x 4 frac e a nbsp V chastkovomu vipadku rivnyannya x 4 p x 2 q x r 0 displaystyle x 4 px 2 qx r 0 nbsp to x 1 x 2 x 3 x 4 0 x 1 x 2 x 1 x 3 x 1 x 4 x 2 x 3 x 2 x 4 x 3 x 4 p displaystyle x 1 x 2 x 3 x 4 0 qquad x 1 x 2 x 1 x 3 x 1 x 4 x 2 x 3 x 2 x 4 x 3 x 4 p nbsp x 1 x 2 x 3 x 1 x 2 x 4 x 1 x 3 x 4 x 2 x 3 x 4 q x 1 x 2 x 3 x 4 r displaystyle x 1 x 2 x 3 x 1 x 2 x 4 x 1 x 3 x 4 x 2 x 3 x 4 q qquad x 1 x 2 x 3 x 4 r nbsp Div takozhred Osnovna teorema algebriDzherelared Kurosh A G Kurs vysshej algebry M Nauka 1968 331 s Otrimano z https uk wikipedia org w index php title Teorema Viyeta amp oldid 40381164