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I know of a theorem in algebra, that every polynomial $p\left(X\right)=a_{0}+a_{1}X+\ldots+a_{n}X^{n}$ that is nonconstant and has real coefficients, admits a factorisation of the form $$ p\left(X\right)=c\left(X-\lambda_{1}\right)\ldots\left(X-\lambda_{m}\right)\left(X^{2}+\alpha_{1}X+\beta_{1}\right)\ldots\left(X^{2}+\alpha_{M}X+\beta_{M}\right), $$ where $m+M\geqslant1$, $c,\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ and $\left(\alpha_{j},\beta_{j}\right)\in\mathbb{R}^{2}$ with $\alpha_{j}<4\beta_{j}$ for $j\in\left\{ 1,\ldots,M\right\} $. My questions are:

1) Is there a name for this theorem ?

2) a) Is there an analogue of this theorem for " polynomials consisting of infinite sums", i.e. for Laurent polynomials ? b) Is there an analogue of this theorem for convergent sums ? (So that they be expressed as a convergent product)

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1 Answer 1

up vote 2 down vote accepted

1) It's a special case of the Fundamental Theorem of Algebra.

2) I think the closest analogue is the Weierstrass Factorization Theorem

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And to 2)a) - do you know if there is analogue for formal power series? –  el le Jun 28 '12 at 18:28

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