Question

We have `f=X^n-4X^2+X+1`.How to determine the rest of the division of `f` to `g=(x-1)^2`?

Answer 1

`r(X)=(n-7)X+6-n`

Explanation:

`f(X)=g(X) q(X)+r(X)`

with `r(X)= aX+b`

and

`f'(X)=g'(X)q(X)+g(X)q'(X)+a`

We know that `g(1)=g'(1)=0` so

`{(1-4+1+1=a+b),(n-2 xx 4+1=a):}`

Solving for `a,b`

`a=n-7`
`b=6-n`

so

`r(X)=aX+b=(n-7)X+6-n`