Can you always use synthetic division for dividing polynomials?
Explain.
No, you cannot always use synthetic division for dividing polynomials. Synthetic division is a simplified method of polynomial long division that can only be used when dividing by a linear factor of the form (x − a), where "a" is a constant. Synthetic division cannot be used when dividing by quadratic or higher degree factors.
Furthermore, synthetic division can only be used when dividing by a factor with a leading coefficient of 1, meaning the coefficient in front of the highest degree term is 1. If the leading coefficient is not 1, polynomial long division must be used instead.
In cases where the divisor is not of the form (x − a), or when the leading coefficient is not 1, polynomial long division is required to accurately divide the polynomials. Polynomial long division involves writing out the division step-by-step, similar to the way we divide numbers, and is a more general and flexible method than synthetic division.