<s>[INST] Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$. I know this question has been answered before, but I have a slightly different different question.
I saw the solution of this question in my book and the author has solved it by substituting $x-1=y$ and then equating the coefficients of $y^2$, $y^1$ and $y^0$  to $A(y+1)^2$, $B(y+1)^1$ and $C$.
My question is why do we have to substitute $x-1=y$ and why can't equate coefficients of $x^2$, $x^1$ and $x^0$ to $A$, $B$ and $C$ without substituting? Thanks in advance.
 [/INST] $$
(x+1)^n = Q(x)(x-1)^3 + ax^2+b x+c
$$
$$ 2^n = a+b+c $$
$$
n(x+1)^{n-1} = Q'(x)(x-1)^3+3Q(x)(x-1)^2 + 2ax+b \Rightarrow n2^{n-1}=2a+b
$$
In the same way
$$
n(n-1)(n-2)2^{n-2} = 2a
$$
etc.
</s>