Linear Algebra, 5th Edition

Appendix E Polynomials

In this appendix, we discuss some useful properties of the polynomials with coefficients from a field. For the definition of a polynomial, refer to Section 1.2. Throughout this appendix, we assume that all polynomials have coefficients from a fixed field F.

Definition.

A polynomial f(x) divides a polynomial g(x) if there exists a polynomial q(x) such that $g(x)=f(x)q(x)$.

Our first result shows that the familiar long division process for polynomials with real coefficients is valid for polynomials with coefficients from an arbitrary field.

Theorem E.1 (The Division Algorithm for Polynomials).

Let f(x) be a polynomial of degree n, and let g(x) be a polynomial of degree $m\ge 0$. Then there exist unique polynomials q(x) and r(