ABSTRACT
In this chapter, we build on high school algebra and develop the algebraic theory of polynomials. In section one we show that under the usual operations of addition and multiplication the collection of all polynomials with coefficients in a field F is a commutative algebra with identity. We define the concepts of greatest common divisor (gcd) and least common multiple (lcm) of two polynomials and make use of the division algorithm (division with remainders) to establish the existence and uniqueness of the gcd and lcm. In section two we prove some general results about roots of polynomials and then specialize to polynomials with coefficients in the fields R and C.
