Question

余式定理...设f(x)以x-1除之,余式为8,以x的平方+x+1除之的余式为7x+16,求x的三次方-1除之的余式为多

Answer 1

设f(x)/(x-1)=q(x)余8,则

根据余式定理,知f(1)=(x-1)q(x)+8=(1-1)q(x)+8=8,

f(x)/(x³-1)=p(x)余n,

则f(x)=(x³-1)p(x)+n,n即为所求,

显然f(1)=(1³-1)p(x)+n=n,

根据前面的推导,知f(1)=8,

所以n=8,

即f(x)除(x³-1)的商是8,