Determine the remainder if f=3x^4-2x^3+x^2+ax-1 is divided by (x-1)^2.

We'll write the division with
reminder:

f(x) = (x-1)^2*C(x) +
R(x)

Since the divisor is a polynomial of second order, the
reminder is a polynomial of first order.

R(x) = mx +
n

We'll calculate f(1):

f(1) =
(1-1)^2*C(1) + R(1)

f(1) =
R(1)

We'll determine f(1), substituting x by 1 in the given
expression of polynomial f:

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

f(1) = 1 + a

R(1) =
m + n

m + n = 1+a (1)

Now,
we'll calculate the first derivative of f(x):

f'(x) =
(3x^4-2x^3+x^2+ax-1)'

f'(x) = 12x^3 - 6x^2 + 2x +
a

f'(x) = [(x-1)^2*C(x) +
R(x)]'

f'(x) = 2(x-1)*C(x) + (x-1)^2*C'(x) +
R'(x)

f'(x) = 2(x-1)*C(x) + (x-1)^2*C'(x) +
m

f'(1) = m

f'(1) = 12 - 6 + 2
+ a

m = 8 + a (2)

We'll
substitute (2) in  (1):

1+a = 8 + a +
n

We'll subtract 8 + a:

n = 1
+ a - 8 - a

We'll eliminate and combine like
terms:

n =
-7

The reminder R(x)
is:

R(x) = (8 + a)x -
7