Determine the reminder of the division of polinomials P/Q P=(x+1)^10+(x+2)^10 Q=(x+1)(x+2)

We'll apply division with
reminder.

P=Q1*Q+R

The degree
of the the reminder has to be smaller than the degree of Q. In our case, Q has the
second degree, so the reminder is a first degree
polynomial.

R=aX+b

First,
let's find out the roots of
Q.

Q=(x+1)(x+2)

(x+1)(x+2) =
0

We'll put each factor as
zero.

(x+1) = 0

We'll add -1
both sides:

x = -1

x+2 =
0

We'll subtract 2 both
sides:

x = -2

Now, we'll
substitute the roots of Q, into the expression of division with
reminder.

P(-1) = Q(-1)*Q1+a*(-1)+b, where
P(-1)=1

But Q(-1)=0,
so:

1=0-a+b

-a+b=1
(1)

P(-2) = Q(-2)*Q2+a*(-2)+b, where
P(-2)=1

1=0-2a+b

-2a+b=1 
(2)

From (1) and (2), we'll
have:

-a+b=-2a+b

We'll
eliminate like
terms:

-a=-2a

a=0, so 0+b=1,
b=1

R=a*X+b

R=0*X+1

R=1