...rFie 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 se va scrie sub forma EMBED Equation.3 . Au loc notatiile EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Exemplu EMBED Equation.3 EMBED Equation.3 Atunci EMBED Equation.3 EMBED Equation.3 I.4. Gradul unui polinomFie EMBED Equation.3 . Se numeste gradul lui EMBED Equation.3 , notat prin EMBED Equation.3 , cel mai mare numar natural n astfel incat EMBED Equation.3 .Exemple 1. Polinomul EMBED Equation.3 are gradul 1 2. Polinomul EMBED Equation.3 are gradul 5 3. Polinomul constant EMBED Equation.3 , unde EMBED Equation.3 ,are gradul 0. Referitor la gradul sumei si produsului a doua polinoame EMBED Equation.3 si EMBED Equation.3 , au loc urmatoarele relatiii EMBED Equation.3 ii EMBED Equation.3 .I.5. Valoarea unui polinom intr-un punctFie EMBED Equation.3 , atunci functia polinomiala asociata polinomului f este EMBED Equatio...
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