## Condition of Roots of Polynomial Equations With Integer Coefficients

A polynomial of degree
$n$
with integer coefficients and coefficient of
$x^n$
equal to 1 can have only integer roots. Moreover if the constant term is
$a_0$
the product of the roots is
$(-1)^na_0$
, so any root must divide this constant term..
To prove the first part let
$x=p/q$
be a root. Substituting this value into the polynomial gives
$a_n\frac{p^n}{q^n} a_{n-1}\frac{p^{n-1}}{q^{n-1}}+...+a_0=0$

Multiply by
$q^{n-1}$
to give
$a_n\frac{p^n}{q} a_{n-1}p^{n-1}+...+a_0q^{n-1}=0$

This is impossible since the first term is a fraction and all the other terms are integers.
For the second part notice that a polynomial with
$n$
roots
$\alpha_1, \; \alpha_2, \; ...\ \alpha_n$
factorises into
$n$
factors
$(x- \alpha_1)(x- \alpha_2)...(x- \alpha_n)$
and when this is expanded the constant term is
$(-1)^n \alpha_1 \alpha_2... \alpha_n$
.
The last part can be used to aid factorisation.
&The only possible rational roots of
$x^4+5x^3+2x^2-10x+6=0$
are
$\pm 1, \; \pm 2, \; \pm 3, \pm 6$
and testing each of these gives only
$x=3$
as an integer root.