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Homeproof of factor theorem using division

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# proof of factor theorem using division

###### Lemma (cf. factor theorem).

Let $R$ be a commutative ring with identity and let $p(x)\in R[x]$ be a polynomial with coefficients in $R$. The element $a\in R$ is a root of $p(x)$ if and only if $(x-a)$ divides $p(x)$.

###### Proof.

Let $p(x)$ be a polynomial in $R[x]$ and let $a$ be an element of $R$.

1. 2. Assume that $a$ is a root of $p(x)$, i.e. $p(a)=0$. Since $x-a$ is a monic polynomial, we can perform the polynomial long division of $p(x)$ by $(x-a)$. Thus, there exist polynomials $q(x)$ and $r(x)$ such that:

$p(x)=(x-a)\cdot q(x)+r(x)$ and the degree of $r(x)$ is less than the degree of $x-a$ (so $r(x)$ is just a constant). Moreover, $0=p(a)=0+r(a)=r(a)=r(x)$. Therefore $p(x)=(x-a)\cdot q(x)$ and $(x-a)$ divides $p(x)$.

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## Mathematics Subject Classification

12D10*no label found*12D05

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