Polynomials/K/Product of linear polynomials and zeroes/Fact/Proof/Exercise

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Show that every polynomial ${}P\in K[X]$, ${}P\neq 0$, can be decomposed as a product

${}P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,$

where ${}\mu _{j}\geq 1$

and ${}Q$ is a polynomial with no roots (no zeroes). Moreover, the different numbers ${}\lambda _{1},\ldots ,\lambda _{k}$ and the exponents ${}\mu _{1},\ldots ,\mu _{k}$ are uniquely determined apart from the order.

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