this question is related to the ones here and here. I am working on the same problem ("how many monic reducible quadratic polynomials are there in $F_p[x]$"), and I'm unsure about the counting argument: there are $p$ polynomials of the form $(x-a)^2$ - ok - but why are there $\frac{p(p-1)}{2}$ of the form $(x-a)(x-b)$? How does this deal with $ab \equiv bc \mod p$ for different $a,b$? Thanks.
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There are $\binom{p}{2}$ ways to choose two distinct elements out of a set $\Bbb F_p$ of $p$ elements. Are you familiar with binomial coefficients, also called combinations? – anon May 22 '20 at 10:44
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@runway44 Yes. My question was with respect to showing that $(x-a)(x-b)$ give distinct quadratics after reduction modulo $p$. – Zarathustra May 22 '20 at 11:20
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They give distinct quadratics because they have different sets ${a,b}$ of zeros. – anon May 22 '20 at 20:14