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 (1)

f = deimpartit g = impartitor q = cat r = rest

Demonstratie:

1.Existenta

f = an Xn + an-1 X n-1 +………+a1 X+a0 Caxi

g= bm Xm +bm-1 X m-1 +………+b1 X +b0 Caxi

grad f = n grad g = m

1.n < m

q = 0

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+1

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 /

u9q2qu

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

(anp / bm) * X np-m )

f = fp+1 +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