Theorem (Root Squaring).  Given the polynomial   of degree n in factored form    with roots  .  Then    is defined by  
    
    .  

is a polynomial of degree n with roots  .  

Proof.

    Consider    and  ,  then their product is

          

          
    
          

which is easily seen to be a polynomial of degree 2n with the roots .  

    However, we choose to use the polynomial  

          

of degree 2n with the roots .  If we replace in the above equation then we obtain

        .

Then we define

          

which is a polynomial of degree n with roots  ,  and the coefficient of the highest power is 1.

(c) John H. Mathews 2005