quadratic polynomial
Học thuậtThân thiện
Definition
- 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
- a polynomial of the second degree