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Abstract Algebra

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Irreducibility and Divisibility of Polynomial Coefficients

ABSALG-EHB2EL

Let:

$P=P(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0\in{\mathbb Z}[x]$

...be a polynomial with $a_0\not=0$ and $a_n\not=0$.

Let $r$, $s$ be integers with greatest common divisor equal $1$ and suppose $P(r/s)=0$.

Which of the following is always true?

A

$r$ divides $a_0$ and $s$ divides $a_n$

B

$s$ divides $a_0$ and $r$ divides $a_n$

C

$r$ divides $a_0$ but $s$ need not divide $a_n$

D

$s$ divides $a_n$ but $r$ need not divide $a_0$