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How about this: Let $k$ be an infinite field, and let $f \in k[x]$. Assume $f(t) = 0$ for all $t \in k$. Assume to the contrary that $f$ is not the zero polynomial. Then $f$ is a polynomial of degree $n \geq 1$. Choose distinct elements $z_1, z_2, z_3, \dots, z_{n+1}$ of $k$. By repeated use of the linear factor theorem, we know that $f(x) = (x-z_1)(x-z_2)(... 4 For me, this is more of a foundational issue than one of secondary education. I would suggest that$\overline {AB}=\overline {BA}$is a matter of equality rather than congruence, given the definition of line segment as the locus of points between$A$and$B$. (Strictly speaking, I suppose, it is the locus of points$C$such that$AC+BC=AB$.) Based on this, ... 2 In axiomatizations of Euclidean plane geometry such as the ones by Hilbert or Tarski, the statement$\overline {AB}\cong\overline {BA}\$ is a postulate. In Tarski's system, this congruence axiom is explicitly called "Reflexivity of Congruence."