Does anyone know a material or even proof of the theorem that says every polynomial of degree n has no more than n REAL roots(real roots, no complex roots)?
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https://math.stackexchange.com/questions/7990/roots-of-a-polynomial-in-an-integral-domain – Rob Arthan Oct 26 '19 at 14:46
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If it is a 'normal' polynomial, that is: of the form $a_n x^n + ... + a_0$ for $a_i \in \mathbb{R}$ it is the fundamental theorem of algebra together with the observation that the real numbers are a subset of the complex numbers. – user388557 Oct 26 '19 at 16:06
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@PeldePinda - actually, since the question is only that $n$ is an upper limit, it is nothing more than the observation that if $P(c) = 0$, then $P(x) = Q(x)(x - c)$ for a polynomial $Q$, whose degree must be $x-1$. By continuing with $Q(x)$ and additional roots, one can have no more than $n$ roots before expressing $P$ as a product of linear factors times a polynomial of degree $0$ -i.e., a constant - which means there are no more roots. The FTA proves the other direction: that the process doesn't end before then with some rootless polynomial. – Paul Sinclair Oct 27 '19 at 02:12