<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] $$(x+1)^n=(x-1)^3Q (x)+ax^2+bx+c $$
for $x=1$, we get
$$\boxed {2^n=a+b+c} $$
differentiating, 
$$n (x+1)^{n-1}=3 (x-1)^2R (x )+2ax+b $$
with $x=1$,
$$\boxed {2^{n-1}=2a+b} $$
differentiating again
$$n (n-1)(x+1)^{n-2}=6 (x-1)S (x)+2a $$
with $x=1$, we find
$$\boxed {a=n (n-1)2^{n-3}}$$
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