No, it's not possible to get the same remainder when a polynomial is divided by two different divisors unless one of the divisors is a factor of the other.
Here's why:
The remainder theorem states that when a polynomial f(x) is divided by (x-a), the remainder is equal to f(a). This means the remainder depends on the specific value 'a' used in the divisor (x-a).
In this case, we are dividing the same polynomial f(a) by two different divisors: (a-1) and (a+1). Since these divisors are not factors of each other (neither is a multiple of the other), the remainder obtained in each case will be different based on the value of 'a' substituted in the respective divisors.
