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#?

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Answer 1 · 2017-04-17T21:35:49

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

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#