## Condition of Roots of Polynomial Equations With Integer Coefficients

\[n\]

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.