determine the reminder of P/Q if P = x^2002 + 2x^2000 - 5x^5 -10x^2 + 2 and Q = (x-1)*x^2 .

We'll write the rule of division with
reminder:

P(x)=Q(x)*C(x)+R(x), where the degree of the
polynomial

R(x)<the degree of the polynomialQ(x).
Because the degree of Q(x) is 3, so the degree of  R(x) is
2.

R(x)=ax^2+bx+c

We'll
calculate the root of Q(x):

(x-1)*x^2 =
0

We'll set each factor as
0:

x-1 = 0

x1 =
1

x^2 = 0

x2=x3 =
0

 The roots of Q(x) are: x1=1,
x2=x3=0.

If we'll substitute x=1 in P(x), we'll
obtain:

P(1)= 1+2-5-10+2

But
P(x)=Q(x)*C(x)+R(x), P(1)=0*C(x)
+a+b+c=a+b+c

a+b+c=-10
(1)

P(0) =
c

P(0)=2

c=2

If
x=0 is a multiple root, this one has to verify the first derivative
,too.

P'(0) =
0

P'(x)=2002x^2001+4000x^1999-25x^4-20x=-2a+b

-2a+b
= 0 (2)

But from (1) and
c=2=>a+b+c=-10=>a+b+2= -10=>a+b=-12
(3)

We'll subtract (2) from
(3):

a+b+2a-b = -12-0

We'll
eliminate like terms:

3a =
-12

We'll divide by
3:

a =
4

We'lll substitute a=4 in
(2):

-2*4+b =
0

b =
8

R(x)=4x^2+8x+2