Polynomials and Matrices

Abstract

Let K be a field. By a polynomial over K we shall mean a formal expression

$$f(t) = {a_n}{t^n} + ... + {a_0}$$

. where t is a “variable”. We have to explain how to form the sum and product of such expressions. Let

$$g(t) = {b_n}{t^m} + ... + {b_0}$$

be another polynomial with b _j ∈ K. If, say, n ≧ m we can write b _j = 0 if j > m,

$${g}t = 0{t^n} + ... + {b_m}{t^m} + ... + {b_0}$$

, and then we can write the sum f + g as

$$(f + g)(t) = ({a_n} + {b_n}){t^n} + ... + ({a_0} + {b_0})$$

.