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

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Answer 1 · 2017-05-10T12:50:53

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

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

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

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

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