Theorem (Separated Real Roots).  If    is a polynomial with real roots that are widely separated in magnitude  

        
then
           for   .   

Exploration

It will suffice to look at the case n=4.


        
  

The coefficients of are:

    

    

    

    

    

(i)  First, we approximate   

    

      

Since ,  each quotients is small  ,  

hence the approximation    is good.

(ii)  Second, we approximate   

      

Divide numerator and denominator by    and get

    
    
    

Since ,  each quotients is small  ,  

hence the approximation    is good.  

(iii)  Third, we approximate

    

Divide numerator and denominator by    and get

      

    

Since ,  each quotients is small  ,  

hence the approximation    is good.  

(iv)  Fourth, we approximate

    

Divide numerator and denominator by    and get

      
      
    
Since ,  each quotients is small  ,  

hence the approximation    is good.

(c) John H. Mathews 2005