Synthetic division cannot be done with a polynomial divisor that is higher than a 1st degree.

Synthetic division is a simplified method of dividing a polynomial by a linear polynomial (a polynomial of degree 1), typically of the form $x - c$. This technique is specifically designed for linear divisors because it relies on the properties of roots and coefficients in the polynomial.

When the divisor is of a higher degree, such as a quadratic polynomial of the form $ax^2 + bx + c$ or any polynomial greater than degree 1, synthetic division cannot be applied. This is due to the fact that synthetic division assumes a linear factor and uses only the coefficients of the dividend polynomial, without accounting for additional terms that might arise from the higher degree of the divisor.

To divide by polynomials of degree higher than 1, one must resort to long division of polynomials, which systematically divides the polynomials and manages all terms according to their degrees, allowing for appropriate subtraction and simplification at each step.

In summary, synthetic division is limited to first-degree divisors because it is specifically designed to manipulate linear factors, and cannot accommodate the complexities involved in higher degree divisors.