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5

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."


2

I really like the idea of discovery fictions because they capture an essential component of how mathematics is best understood and communicated. Almost all ideas in mathematics follow simply and inexorably from previous ideas, and understanding any mathematical discipline consists almost entirely of figuring out how you could have developed these ideas on ...


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