已知f(x)=x^3+2x^2-x+3,g(x-1)=f(x)及h(x+1)=g(x+3),則f(x)+xh(x)除以x+1之餘?

f(x)=x^3+2x^2-x+3,
g(x-1)=f(x)   ==>   g(x) = f(x+1)
h(x+1)=g(x+3)  ==> h(x) = g(x+2) = f(x+3)
∴ f(x)+xh(x) = f(x)+x．f(x+3)
一多項式 p(x) 除以 x-a 的餘式是 p(a).
故 f(x) + x f(x+3) 除以 x+1 其餘式為
   f(-1) + (-1)f(-1+3) = f(-1)-f(2)
      = (-1+2+1+3) - (8+8-2+3)
      = 5 - 17 = -12