RationPolynomen

RationPolynomen are polynomials whose coefficients belong to the field of rational numbers, denoted Q. In mathematics they are elements of the polynomial ring Q[x]. A typical RationPolynomen has the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 with a_i ∈ Q. The degree is n if a_n ≠ 0; the zero polynomial has no meaningful degree in the usual sense.

RationPolynomen are closed under addition, subtraction, and multiplication, and they form a commutative ring with identity. Since Q is a field, Q[x] is an integral domain and, in fact, a principal ideal domain and a unique factorization domain. Rational coefficient polynomials can be factored into irreducible polynomials over Q, with the Fundamental Theorem of Algebra assuring factorizations over C; over Q, irreducible polynomials play a key role in number theory. The Rational Root Theorem characterizes rational roots of such polynomials with integer coefficients, and scaling arguments allow analyzing roots over Q.

Examples include p(x) = (3/4)x^2 − 5x + 2 and q(x) = x^3 − (1/2)x + 7/3.

Polynomials with rational coefficients are often contrasted with polynomials with integer coefficients; Z[x] ⊆ Q[x], and many algorithms for exact arithmetic operate in Q[x]. Rational polynomials serve in symbolic computation, interpolation, generating functions, and as building blocks in algebraic number theory. They also underpin the study of rational functions, which are ratios of polynomials.

Notes on terminology: in some languages or texts the term "RationPolynomen" is used informally to refer to polynomials over Q; the standard term is "polynomials with rational coefficients" or "polynomials over the field of rational numbers".