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?

1 Answer
Apr 17, 2017

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