由高中代数(甲种本)第三册第19页的定理:“复系数一元n次方程在复数集C中有且仅有n个根(k个重根算作k个根)”,可以引出推论: 使复系数多项式f(x)=a_0x~n+a_1x~(n-1)+…+a_n之值为零的相异x值如多于n个,则a_0=a_1=a_2=…=a_n=0(即f(x)≡0)。(*) 推论(*)易由反证法证明。因为若a_0≠0,则由定理可知,满足f(x)=0的不同x值最多有n个,这与己知使f(x)的值为零的不同x值多于n个相矛盾。所以,a_0=0。同 Theorem of high school algebra (a typebook) Vol. 3, page 19: “The complex coefficient unary n-order equations have only n roots in the complex set C (k number of roots counted as k roots)”, which can be derived Corollary: If there are more than n distinct x values ​​such that the value of the complex coefficient polynomial f(x)=a_0x~n+a_1x~(n-1)+...+a_n is zero, then a_0=a_1=a_2=...= A_n=0 (ie, f(x) ≡ 0). (*) Inference (*) is easily proved by counter-evidence. Since if a_0≠0, the theorem shows that there are at most n different x values ​​that satisfy f(x)=0, which is inconsistent with the fact that it is known that making the value of f(x) zero for different x values ​​is more than n . So a_0=0. with