So in Rothstein - Trager's Method of evaluating logarithmic part they need resultant of two polynomial as shown in the image. My question is that how do they calculate the resultant without knowing the roots of these two polynomials? And if they know the roots then they can directly factorize the polynomials instead of going like this.
The resultant is defined as the determinant of the Sylvester matrix. Efficient computation is effected by the sub-resultant algorithm.