Definition

1. Noun:
   • A quadratic polynomial is a polynomial where the highest power (exponent) of the variable is 2. Its standard form is ( ax^2 + bx + c ), where ( a ), (b), and (c) are constants (coefficients) and ( a \neq 0 ).

Usage

• The term quadratic polynomial is used in algebra and mathematics to describe a specific type of polynomial function. It is the core object of study in quadratic equations and functions.
• It is often simply called a quadratic.

Examples

• Noun:
  • The expression ( 3x^2 - 5x + 2 ) is a quadratic polynomial.
  • To find the roots, we set the quadratic polynomial equal to zero.
  • The graph of any quadratic polynomial is a parabola.

Advanced Usage

• "To factor a quadratic polynomial": To express it as the product of two linear polynomials (binomials).
  • We can factor the quadratic polynomial ( x^2 - 4 ) into ( (x-2)(x+2) ).
• "Standard form of a quadratic polynomial": The conventional arrangement ( ax^2 + bx + c ).
  • Rewrite the expression ( 5 + 2x^2 - 3x ) in the standard form of a quadratic polynomial.

Variants and Related Words

• Quadratic (n): A common short form for "quadratic polynomial" or "quadratic expression".
  • Solve the following quadratic: ( 2x^2 + 7x - 15 = 0 ).
• Quadratic function (n): A function defined by a quadratic polynomial, typically written as ( f(x) = ax^2 + bx + c ).
• Quadratic equation (n): An equation that sets a quadratic polynomial equal to zero (( ax^2 + bx + c = 0 )).

Synonyms

• Second-degree polynomial: A direct synonym emphasizing the highest exponent is 2.

Related Terms and Concepts

• Coefficient (n): The constant numbers multiplying the variable, specifically ( a ), ( b ), and ( c ) in a quadratic polynomial.
• Parabola (n): The U-shaped curve that is the graph of a quadratic function.
• Discriminant (n): The expression ( b^2 - 4ac ) derived from the coefficients of a quadratic polynomial, which determines the nature of its roots.

Noun

1. a polynomial of the second degree