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.

Add comment

Security code
Refresh