SOLUTION: Find the remainder when (x+1)^n is divided by (x-1)^3

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Question 1049427: Find the remainder when (x+1)^n is divided by (x-1)^3
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Q%28x%29= quotient
R%28x%29=ax%5E3%2Bbx%2Bc= remainder
%28x%2B1%29%5En=Q%28x%29%2A%28x-1%29%5E3%2Bax%5E2%2Bbx%2Bc
Let y be
y=x-1 <----> x=y%2B1 <---> x%2B1=y%2B2
Now,
%28y%2B2%29%5En=Q%28y%2B1%29%2Ay%5E3%2Ba%28y%2B1%29%5E2%2Bb%28y%2B1%29%2Bc

Q%28x%29%2Ay%5E3 has only terms if degree 3 and greater than 3,
and those are not our problem.
If that equation is true for every y ,as it should,
The terms of degree less than 3 must be equivalent, so

must be true for every value of y .
That means that we have to solve the system of equations
system%28a=n%28n-1%29%2A2%5E%28n-2%29%2F2%2C2a%2Bb=n%2A2%5E%28n-1%29%2Ca%2Bb%2Bc=2%5En%29 .
highlight%28a=n%28n-1%29%2A2%5E%28n-3%29%29
b=n%2A2%5E%28n-1%29-2a
b=n%2A2%5E%28n-1%29-2%2An%28n-1%29%2A2%5E%28n-3%29
b=n%2A2%5E%28n-2%29%2A2-n%28n-1%29%2A2%5E%28n-2%29
b=n%2A2%5E%28n-2%29%2A%282-n%2B1%29
b=n%2A2%5E%28n-2%29%2A%283-n%29
highlight%28b=-n%28n-3%29%2A2%5E%28n-2%29%29
c=2%5En-a-b
c=2%5En-n%28n-1%29%2A2%5E%28n-3%29%2Bn%28n-3%29%2A2%5E%28n-2%29
c=2%5E%28n-3%29%2A2%5E3-n%28n-1%29%2A2%5E%28n-3%29%2Bn%28n-3%29%2A2%5E%28n-2%29%2A2
c=2%5E%28n-3%29%2A%282%5E3%2Bn%2A%282%28n-3%29-%28n-1%29%29%29
c=2%5E%28n-3%29%2A%288%2Bn%282n-6-n%2B1%29%29
highlight%28c=2%5E%28n-3%29%288-n%28n-5%29%29%29