Section 3.2
Division of polynomials

For the polynomial ring R[X], where R = Q, R, C, or Z/pZ, with p prime, we introduce - similar to the integer case - division with remainder. In the integer case this involved comparing integers by their size. For polynomials the degree is the appropriate measure.

Definition

Let a = a₀ + a₁X + ··· + a_nXⁿ be a polynomial in R[X] with a_n 0. We call

• a_nXⁿ the leading term and a_n the leading coefficient of a;
• n the degree of the polynomial a.

We denote the degree of the polynomial a by deg(a).

 >>> Examples Gapplet Exercise

If all the coefficients of a polynomial a are equal to 0, then a = 0 (the zero polynomial). It is practical to define the degree of the zero polynomial to be -1.

A polynomial of degree 1 is also called a linear polynomial. A polynomial is said to be monic if its leading coefficient is equal to 1.

If the nonzero polynomial a has leading coefficient a_n and the nonzero polynomial b has leading coefficient b_m, then the leading coefficient of ab is a_nb_m as follows from the definition of the product. In that case