高一數學:多項式 麻煩各位?

假設f(x)=(x－1)(x－2)(x－3)(x－4)(x－5)(x－6),試求f(f(x)))除以f(x)之餘式
SoL
Set g(x)=f(f(x))
degf(x)=6
Set g(x)=h(x)f(x)+ax^5+bx^4+cx^3+dx^2+ex+k
p(x)=ax^5+bx^4+cx^3+dx^2+ex+k
g(x)=h(x)f(x)+p(x)
g(1)=f(f(1))=f(0)=720=p(1)
g(2)=f(f(2))=f(0)=720=p(2)
g(3)=f(f(3))=f(0)=720=p(3)
g(4)=f(f(4))=f(0)=720=p(4)
g(5)=f(f(5))=f(0)=720=p(5)
g(6)=f(f(6))=f(0)=720=p(6)
degp(x)=5
or
Set g(x)=f(f(x))
degf(x)=6
Set g(x)=f(f(x))=h(x)f(x)+a(x－1)(x－2)(x－3)(x－4)(x－5)+b(x－1)(x－2)(x－3)(x－4)+c(x－1)(x－2)(x－3)+d(x－1)(x－2)+e(x－1)+k
g(1)=h(1)f(1)+k=k=f(f(1))=f(0)=720
g(x)=f(f(x))=h(x)f(x)+a(x－1)(x－2)(x－3)(x－4)(x－5)+b(x－1)(x－2)(x－3)(x－4)+c(x－1)(x－2)(x－3)+d(x－1)(x－2)+e(x－1)+720
g(2)=h(2)f(2)+e+720=f(f(2))=f(0)=720
e=0
g(x)=f(f(x))=h(x)f(x)+a(x－1)(x－2)(x－3)(x－4)(x－5)+b(x－1)(x－2)(x－3)(x－4)+c(x－1)(x－2)(x－3)+d(x－1)(x－2)+720
g(3)=h(3)f(3)+2d+720=f(f(3))=f(0)=720
d=0
g(x)=f(f(x))=h(x)f(x)+a(x－1)(x－2)(x－3)(x－4)(x－5)+b(x－1)(x－2)(x－3)(x－4)+c(x－1)(x－2)(x－3)+720
g(4)=h(4)f(4)+6c+720=f(f(4))=f(0)=720
c=0
g(x)=f(f(x))=h(x)f(x)+a(x－1)(x－2)(x－3)(x－4)(x－5)+b(x－1)(x－2)(x－3)(x－4)+720
g(5)=h(5)f(5)+24b+720=f(f(5))=f(0)=720
b=0
g(x)=f(f(x))=h(x)f(x)+a(x－1)(x－2)(x－3)(x－4)(x－5)+720
g(6)=h(6)f(6)+120a720=f(f(6))=f(0)=720
a=0
g(x)=f(f(x))=h(x)f(x)+720