It seems there is a great confusion of terms in this area.

What is commonly called synthetic division is just one type of synthetic division, division of a polynomial by , for which it seems the best name is **Ruffini’s rule**.

Horner’s rule, or **Horner’s form**, is a polynomial in the form:

Horner’s method is a way of finding roots of a polynomial, by repeatedly reducing the equation by the integer part of a root, and multiplying the coefficients by factors of 10 to obtain further digits.

These are related but separate things.

E. J. Barbeau’s *Polynomials – a problem book* describes evaluating a polynomial by ‘constructing Horner’s table’ (i.e. plugging in e.g. to the Horner form) as ‘Horner’s method’. And then shows that this is an identical process to dividing the polynomial by . I suppose the confusion comes from these 2 procedures being identical. Although neither is apparently Horner’s method. He does division by polynomials of a higher degree with “*Horner’s Method of Synthetic Division* which can be regarded as a generalization of his method for division by a binomial “.

~~++++~~(He also adapts Horner’s table to convert powers and polynomials to falling (factorial) powers and polynomials – see below.)

Dividing a polynomial by with Ruffini’s rule either results in a remainder of , in which case is a root, or a remainder .

The **polynomial remainder theorem** states:

**If a polynomial is divided by , then the remainder is a constant given by .**

Barbeau’s table used to convert to falling powers:

**references**

Mathews, Walker – Mathematical Methods of Physics, 1971, p361-3