POLYNOMIALS Gabriel

for some coefficients ci. Admittedly this definition is not quite all there in that it doesn’t say what the coefficients are. Generally, we take our coefficients in some field. A field is a system of objects with two operations (generally called “addition” and “multiplication”), both of which are commutative and associative, have identities, and are related by the distributive law; also, every element has an additive inverse, and everything except the additive identity has a multiplicative inverse. Examples of familiar fields are the rational numbers Q, the reals R, and the complex numbers C, though plenty of other examples exist, both finite and infinite. We let F [x] denote the set of all polynomials “over” (with coefficients in) the field F . Unless otherwise stated, don’t worry about what field we’re working over. A few more terms should be defined before we proceed. First, if f(x) = ∑n i=0 cix i with cn 6= 0, we say n is the degree of the polynomial, written deg f . Thus the degree simply means the highest power of x that occurs. The zero polynomial f(x) = 0 is usually held to have no degree, though some folks like to say it has degree −∞; when it is convenient we will assume it has degree less than any other polynomial. The polynomials of degree 0, together with the zero polynomial, are called constant polynomials. If f(x) = ∑n i=0 cix i with cn 6= 0, then cn is the leading coefficient and c0 is the constant term. A monic polynomial is one with leading coefficient 1. For convenience, we’ll usually that we write our polynomials so that cn 6= 0. But when that makes life annoying, we can equivalently say that ci = 0 for i > deg f (and for i < 0, while we’re at it). Polynomials are useful because we can look at them either as purely algebraic objects or as functions of the variable x. For now let’s get some algebraic properties down. We can’t write any equations until we know what equality means, so let’s write f = g if the coefficients match, i.e. in the form f(x) = ∑n i=0 cix , g(x) = ∑m j=0 djx j we have n = m and ci = di for each i. Now we can clearly add two polynomials, by adding like terms: if f(x) = ∑n i=0 cix i and g(x) = ∑m j=0 djx j , then