matematica - impartirea polinoamelor



Impartirea polinoamelor


1.Teorema impartirii cu rest


Fiind date doua polinoame oarecare cu coeficienti complecsi f si g cu g<>0, atunci exista doua plinoame cu coeficienti complecsi q si r a .i.




f = gq+r unde grad r < grad g (1)


In plus polinoamele q si r sunt unice satisfacand proprietatea



f = deimpartit

g impartitor

q cat

r = rest



Demonstratie


1.Existenta


f = an Xn + an-1 X n-1 +...+a1 X+a0 C[x]


g= bm Xm +bm-1 X m-1 +...+b1 X +b0 C[x]


grad f = n

grad g = m


1.n < m


q


f=0*g+f


2.n >= m


an / bm

an Xn / bm Xm

q1= (an / bm) * X n-m


f (an / bm) * X n-m ) *g + f1        (1)


grad f1 = n1 <grad f = n


an Xn + an-1 X n-1 +...+a1 X+a0/ : bmXm


f1= an1 Xn1 + an1-1 X n1-1 +...+a11 X+a01


Daca gr. f1 =n1


i) gr f1 < gr g STOP


ii) daca gr f1 >= gr g


f1 (an1 / bm) * X n1-m ) *g + f2        (2)


gr f2=n2 <n1 < n


i)       gr n2<m STOP

ii)            gr n2>=m


f2 (an2 / bm) * X n2-m ) *g + f3        (3)



p pasi

p


fp (anp / bm) * X np-m ) *g + fp+1        (p+1)


gr f p+1<m      STOP







f1= f ( (an / bm) * X n-m ) *g /


f2= f1 (an1 / bm) * X n1-m ) *g       /


f3= f2- ( (an2 / bm) * X n2-m ) *g       / +


/


f p+1= f p (anp / bm) * X np-m) *g       /






f p+1 = f - g ((an / bm) * X n-m + (an1 / bm) * X n1-m +...+


(anp / bm) * X np-m )


f = fp +g ((an / bm) * X n-m + (an1 / bm) * X n1-m +...+


(anp / bm) * X np-m )


q = ((an / bm) * X n-m + (an1 / bm) * X n1-m +...+


(anp / bm) * X np-m )


r = f p+1


Gr f p+1< m