Question

We have `f,ginRR[X];f=(X-1)^n-X^n+1;g=X^2-3X+2`.How to find the rest of dividing `f` to `g`?

Answer 1

`r(x) = (2-2^n)x+2^n-2`

Explanation:

For degree consistence in `f(x) = q(x)g(x)+r(x)`

If `deg(f)=n-1` and `deg(g)=2` then

`deg(q)=n-1-2=n-3` and `deg(r) = 1`

so

`r(x) = ax+b` and

`g(x)=(x-1)(x-2)`

so

`f(x)=q(x)(x-1)(x-2)+a x + b`

Now we have

`f(1)=a cdot 1 + b = 0`
`f(2)= a cdot 2 + b =1-2^n+1 = 2-2^n`

Solving

`{(a cdot 1 + b = 0),(a cdot 2 + b = 2-2^n):}`

for `a,b` we have `a = 2-2^n` and `b=2^n-2`

and finally

`r(x) = (2-2^n)x+2^n-2`