The algebra of real polynomials

Polynomials

A polynomial with real coefficients in the variable $x$ is an expression of the following form, where $n$ is a natural number or zero and the coefficients $a_0, a_1, \ldots, a_n$ are real numbers:

Each summand $a_i x^i$ is called a term, and $a_0$ is the constant term. Each polynomial defines a function: the one that assigns to each real number $x$ the number $P(x)$ obtained by replacing the variable with $x$ in the expression. The functions obtained in this way, called polynomial functions, are the simplest algebraic functions, introduced in the lesson on algebraic functions.

Two polynomials are equal when they have the same coefficients, power by power, once the powers missing from either of them have been filled in with zero coefficients; thus $1 + 0x + 3x^2$ and $1 + 3x^2$ are the same polynomial. The polynomial whose coefficients are all zero is called the zero polynomial.

The degree of a nonzero polynomial is the largest exponent whose coefficient is different from zero; that coefficient is called the leading coefficient. Thus $P(x) = 3 - x + 5x^4$ has degree $4$ and leading coefficient $5$, and one writes $\deg(P) = 4$. The zero polynomial has no degree, because it has no coefficient different from zero. Since the degree and the leading coefficient are read off the coefficients, and two equal polynomials have the same coefficients, both are well defined.

Now, a polynomial is almost always used through the function it defines, and one may ask whether two different polynomials could define the same function; if so, the degree of a polynomial function would not be well defined. The identity principle, which is proved in the section on the number of roots, answers no: two polynomials define the same function if and only if they are equal. Until then, everything said about polynomials, and in particular about their degree, refers to their coefficients, so that no proof prior to that principle depends on it. Afterwards, each polynomial can be identified with its function, and comparing coefficients in an equality between polynomial functions is justified, as in solved problem 5.

Classification

Polynomials are classified, in the first place, by their degree:

1. Constants: those of degree $0$, of the form $P(x) = c$ with $c \neq 0$, together with the zero polynomial.
2. Linear, of degree $1$: $P(x) = a_0 + a_1 x$, with $a_1 \neq 0$.
3. Quadratic, of degree $2$, and cubic, of degree $3$; those of higher degrees are named by their number: of degree $4$, of degree $5$, and so on.

By its number of nonzero terms, a polynomial is called a monomial if it has one, a binomial if it has two and a trinomial if it has three.