Condition of Roots of Polynomial Equations With Integer Coefficients

A polynomial of degree  
  with integer coefficients and coefficient of  
  equal to 1 can have only integer roots. Moreover if the constant term is  
  the product of the roots is  
, so any root must divide this constant term..
To prove the first part let  
  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  
 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  
\[\alpha_1, \; \alpha_2, \; ...\ \alpha_n\]
  factorises into  
\[(x- \alpha_1)(x- \alpha_2)...(x- \alpha_n)\]
  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  
\[\pm 1, \; \pm 2, \; \pm 3, \pm 6\]
  and testing each of these gives only  
  as an integer root.

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