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define polynomial
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A polynomial (in one variable x) over a coefficient ring or field R is an expression of the form
a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0,
where n is a nonnegative integer, each a_i ∈ R, and a_n ≠ 0. The terms a_i x^i are called monomials and the a_i are the coefficients.
Key points:
- Degree: n is the degree of the polynomial. A nonzero constant has degree 0. The zero polynomial has no well-defined finite degree (it is sometimes assigned degree −∞).
- Polynomial ring: The set of all such polynomials with coefficients in R is denoted R[x].
- Multivariable: A polynomial in variables x, y, ... is a finite sum of terms c x^i y^j ... with nonnegative integer exponents.
- Operations: Polynomials are closed under addition, subtraction and multiplication. Division by a nonconstant polynomial requires quotient and remainder (division algorithm) when R is a field.
- Roots/factors: A value r (in a larger field) is a root if substituting x = r gives zero; then (x − r) is a factor (over a field).
Examples: 2x^2 − 3x + 1, 7 (constant polynomial), 0 (zero polynomial), x^3y − 4xy^2 (multivariable).
a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0,
where n is a nonnegative integer, each a_i ∈ R, and a_n ≠ 0. The terms a_i x^i are called monomials and the a_i are the coefficients.
Key points:
- Degree: n is the degree of the polynomial. A nonzero constant has degree 0. The zero polynomial has no well-defined finite degree (it is sometimes assigned degree −∞).
- Polynomial ring: The set of all such polynomials with coefficients in R is denoted R[x].
- Multivariable: A polynomial in variables x, y, ... is a finite sum of terms c x^i y^j ... with nonnegative integer exponents.
- Operations: Polynomials are closed under addition, subtraction and multiplication. Division by a nonconstant polynomial requires quotient and remainder (division algorithm) when R is a field.
- Roots/factors: A value r (in a larger field) is a root if substituting x = r gives zero; then (x − r) is a factor (over a field).
Examples: 2x^2 − 3x + 1, 7 (constant polynomial), 0 (zero polynomial), x^3y − 4xy^2 (multivariable).
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