YLL討論網

X^2000 除以 X^3+2x^2+2x+1 ,求餘式?
X^3+2x^2+2x+1=(x+1)(x^2+x+1)
令f(x)=x^2000
f(-1)=1
所以f(x)=(x+1)Q(x)+1
f(x)=x^2000=(x^3)^666*(x^2)
令x^3-1=0代入
=>x^2000=(x^3-1)K(x)+x^2=(x^3-1)K(x)+(x^2+x+1)+(-x-1)
f(x)=(x+1)(x^2+x+1)P(x)+a(x^2+x+1)+(-x-1)
f(-1)=a(1-1+1)-(-1)-1=1
=>a=1
f(x)=(x+1)(x^2+x+1)P(x)+(x^2+x+1)+(-x-1)
=(x+1)(x^2+x+1)P(x)+x^2
所以X^2000 除以 X^3+2x^2+2x+1的餘式為x^2

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