degree of a polynomial

Thân thiện

A student writes the degree of a polynomial on a chalkboard.

Definition

1. Noun:
   • The degree of the term in the polynomial that has the highest degree: In mathematics, the degree of a polynomial is the highest power (exponent) of the variable that appears in any term of the polynomial when it is expressed in its standard form (with terms ordered from highest to lowest degree). For a polynomial in a single variable, it is the largest exponent on that variable.

Usage Examples

• Noun:
  • The degree of the polynomial 3x^4 + 2x^2 - x + 7 is 4.
  • To find the degree of a polynomial, first combine like terms and then identify the term with the largest exponent.

Advanced Usage

• Degree of a Zero Polynomial: The degree of the zero polynomial (e.g., 0 or 0x^2) is typically considered undefined or, in some contexts, said to be -∞ (negative infinity) to preserve certain mathematical properties.

  • In advanced algebra, the degree of the zero polynomial is often left undefined.

• Degree of a Multivariate Polynomial: For polynomials with more than one variable, the degree is the highest sum of the exponents in any single term.

  • The degree of the polynomial 2x^2y^3 + xy is 5, because the sum of the exponents in the term 2x^2y^3 is 2 + 3 = 5.

Variants and Related Words

• Polynomial (n): An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

  • x^2 - 4x + 4 is a polynomial.

• Leading Term (n): The term in a polynomial with the highest degree.

  • In the polynomial 5x^3 - x + 1, the leading term is 5x^3.

• Leading Coefficient (n): The coefficient of the leading term.

  • For the polynomial -2x^5 + 3x, the leading coefficient is -2.

Synonyms

• Order of a polynomial: A less common but sometimes used synonym, especially in older texts or specific contexts.

Related Phrases

• To find the degree of a polynomial: The common instructional phrase for the action.

  • The first step in analyzing the polynomial is to find its degree.

• Polynomial of degree n: A common descriptive phrase.

  • A quadratic equation involves a polynomial of degree 2.

Học thuật

Noun

1. the degree of the term in the polynomial that has the highest degree