In the following notes https://people.maths.bris.ac.uk/~mazag/nt/lecture6.pdf the proof of Lagrange's theorem (Theorem 1.2), I am unsure as to how the final line is produced. Namely that the number of zeroes of $f(x)\mod p$ is at most the sum of the number of zeroes of its factorization which in this case we have $g(x)$ and $x-a$.
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1A product is $0$ modulo $p$ if and only if one of the factors is $0$ modulo $p$. Since $f(x)\equiv g(x)(x-a)\pmod{p}$, if $f(b)\equiv 0\pmod{p}$, then $g(b)(b-a)\equiv 0\pmod{p}$, so $g(b)\equiv 0\pmod{p}$ or $b-a\equiv 0\pmod{p}$ (since $p$ divides a product if and only if it divides at least one of the factors). – Arturo Magidin Oct 31 '20 at 00:02
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How do we know these are the only solutions? – someone1 Oct 31 '20 at 01:18
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Huh? Because any solution will necessarily force either $g(b)$ to be $0$, or $(b-a)$ to be $0$ (modulo $p$). If $g(b)\not\equiv 0$ and $(b-a)\not\equiv 0$, then $g(b)(b-a)\not\equiv 0$, so $f(b)\not\equiv 0$. What is the problem? – Arturo Magidin Oct 31 '20 at 01:55
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You may find it helpful to read the ]proof of the BiFactor Theorem, and its linked post(s). – Bill Dubuque Oct 31 '20 at 09:25