Which is the reminder of the division f/g if f=x^5+3x^3-2x^2+x+1 and g=x^3-x ?

First , we'll write the rule of division with
reminder:

f(x)=g(x)*C(x)+R(x), where the degree of the
polynomial

R(x) < the degree of the polynomial
g(x).

Because the degree of g(x) is 3, the degree of 
R(x) will
be 2.

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

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

x^3-x =
0

We'll factorize by
x:

x(x^2-1) = 0

x(x-1)(x+1) =
0

We'll set each factor as
0:

x1 =
0

x-1 =
0

x2 =
1

x+1 =
0

x3 =
-1

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

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

f(1)= 1^5 + 3*1^3 - 2*1^2 + 1 +
1

f(1) = 1 + 3 - 2 + 1 +
1

f(1) = 4

But
f(x)=g(x)*C(x)+R(x),

f(1)=0*C(1) +a+b+c =
a+b+c

a+b+c=4
(1)

Now, we'll substitute x=0 in f(x)
and we'll obtain:

f(0) =
c

f(0)=1

c=1

We'll
substitute x by -1:

f(-1) =
-1-3-2-1+1

f(-1) =
-6

But

f(-1) = 0*C(-1)  +
(a-b+c)

f(-1) =
(a-b+c)

(a-b+c) = -6
(2)

We'll substitute c = 1 in (1) and
(2):

a+b+1 = 4

a+b = 3
(3)

a-b+1 = -6

a-b = -7
(4)

We'll add (3) and
(4):

a+b+a-b = 3-7

2a =
-4

We'll divide by
2:

a =
-2

We'll sbstitute a = -2 in
(3):

a+b = 3

-2 + b =
3

b =
5

Thereminder R(x)
is:

R(x) = -2x^2 + 5x +
1