.... Multimea polinoamelor cu coeficienti reali .13III. Multtimea polinoamelor cucoeficienti intregi si rationali 14IV. Aplicatii ..15IV.1. Probleme rezolvate 15IV.2. Probleme propuse ..19Polinoame cu coeficienti complecsiI. Multimea polinoamelor cu coeficienti complecsiI.1.Definirea polinoamelorFie CiXs multimea sirurilorinfinite de numerecomplexe EMBED Equation.3 , care au numai un numar finit de termeni ai,nenuli, adica exista un numar natural m, astfel incat ai0, pentru orice im.De exemplu, sirurile EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 sunt siruri infinite care au un numar finit de termeni nenuli. irul g are 3 termeni nenuli, iar h are 4 termeni nenuli. Deci aceste siruri sunt elemente din multimea CiXs.I.2. Adunarea si inmultirea polinoamelorDefinim pe multimea CiXs doua operatii algebrice adunarea si inmultirea.Adunarea polinoamelorFie EMBED Equation.3 , EMBED Equation.3 doua elemente din multimea CiXs atunci definim EMBED Equation.3 , EMBED Equation.3 Proprietatile adunarii polinoamelorCiXs, se numeste grup abelianAsociativitatea EMBED Equation.3 , EMBED Equation.3 CiXs Intr-adevar, daca EMBED Equation.3 , EMBED Equation.3 si EMBED Equation.3 atunci avem EMBED Equation.3 si deci EMBED Equation.3 .Analog, obtinem ca EMBED Equation.3 . Cum adunarea numerelor este asociativa, avem EMBED Equation.3 , pentru orice EMBED Equation.3 .Comutativitatea EMBED Equation.3 , EMBED Equation.3 CiXsIntr-adevar, daca EMBED Equation.3 si EMBED Equation.3 , avem EMBED Equation.3 , EMBED Equation.3 Cum adunarea numerelor complexe este comutativa, avem EMBED Equation.3 pentru orice EMBED Equation.3 . Deci EMBED Equation.3 .Element neutru Polinomul constant 00,0,0, este element neutru pentru adunarea polinoamelor, in sensul ca oricare ar fi EMBED Equation.3 CiXs,avem EMBED Equation.3 Elemente inversabileOrice polinom are un opus, adica oricare ar fi EMBED Equation.3 CiXs, exista un polinom, notat EMBED Equation.3 , astfel incat EMBED Equation.3 De exemplu, daca EMBED Equation.3 este un polinom, atunci opusul sau este EMBED Equation.3 EMBED Equation.3 Inmultirea polinoamelorFie EMBED Equation.3 , EMBED Equation.3 Atunci definim EMBED Equation.3 ck EMBED Equation.3 Proprietatile inmultiriiAsociativitateaOricare ar fi EMBED Equation.3 CiXs, avem EMBED Equation.3 ComutativitateaOricare ar fi EMBED Equation.3 CiXs,avem EMBED Equation.3 Intr-adevar, daca EMBED Equation.3 , EMBED Equation.3 , atunci notand EMBED Equation.3 si EMBED Equation.3 , avem EMBED Equation.3 si EMBED Equation.3 . Cum adunarea si inmultirea numerelor complexe sunt comutative si asociative, avem crdr, pentru orice EMBED Equation.3 . Deci EMBED Equation.3 .Element neutruPolinomul 11,0,0, este element neutru pentru inmultirea polinoamelor, adica oricare ar fi EMBED Equation.3 CiXs,avem EMBED Equation.3 Elemente inversabile EMBED Equation.3 CiXs este inversabil daca exista EMBED Equation.3 ,a.i. EMBED Equation.3 Singurele polinoame inversabile sunt cele constante nenule EMBED Equation.3 , a0.DistributivitateaOricare ar fi polinoamele EMBED Equation.3 CiXs,are loc relatia EMBED Equation.3 1.3. Forma algebrica a polinoamelorNotatia EMBED Equation.3 introdusa pentru polinoame nu este prea comoda in operatiile cu polinoame. De aceea vom folosi alta scriere.Daca consideram EMBED Equation.3 , atunci EMBED Equation.3 ...
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