What is polynomial ax^3+bx^2+cx+d if divided by (x-1),(x+1),(x+2), the reminder is always 3?

We'll note the polynomial P(X)
= aX^3+bX^2+cX+d.

We'll write the reminder theorem, when
P(x) is divided by
(X-1):

P(1)=3

We'll write the
reminder theorem, when P(x) is divided by
(X+1):

P(-1)=3

We'll write the
reminder theorem, when P(x) is divided by
(X+2):

P(-2)=3

From these
facts, we notice that the reminder of the division of   P(x) to the product of
polynomials (X-1)(X+1)(X+2) is also 3.

We'll write the
reminder theorem:

aX^3+bX^2+cX+d=(X-1)(X+1)(X+2) +
3

We'll remove the
brackets:

aX^3+bX^2+cX+d=(X^2-1)(X+2)
+3

aX^3+bX^2+cX+d = X^3 + 2X^2-X -2 +
3

We'll combine like terms and we'll
get:

aX^3 + bX^2 + cX + d = X^3 + 2X^2 - X +
1

P(X) = X^3 + 2X^2 - X +
1