﻿

In mathematics, a quadratic polynomial is a polynomial whose degree is 2. A quadratic polynomial with three terms is called a quadratic trinomial. Some examples of quadratic polynomials are "ax"2 + "bx" + "c", 2"x"2 − "y"2, and "xy" + "xz" + "yz".=Coefficients=The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring.=Degree=When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant. =Variables=

The one-variable case

If the polynomial is a polynomial in one variable, it determines a quadratic function in one variable. An example is given by "f"("x") = "x"2 + "x" − 2;. The graph of such a function is a parabola (in degenerate cases a line), and its zeroes can be found by solving the quadratic equation "f"("x") = 0.

There are 3 main forms :
* general form, $f\left(x\right) = a_2 x^2 + a_1 x + a_0 ,$.
* logistic form, $f_r\left(x\right) = r x \left( 1-x \right) ,$, used to study 1D discrete dynamics,
* monic and centered form, $f_c\left(x\right) = x^2 +c,$, used to study complex dynamics.

The general case

In the general case, a quadratic polynomial in "n" variables "x"1, ..., "x""n" can be written in the form:$sum_\left\{i, j = 1\right\}^\left\{n\right\} Q_\left\{i,j\right\} x_i x_j + sum_\left\{i = 1\right\}^\left\{n\right\} P_i x_i + R$where "Q" is a symmetric "n"-dimensional matrix, "P" is an "n"-dimensional vector, and "R" a constant.=Roots=The zeroes of a quadratic polynomial form a quadric. The conic sections, such as ellipse and hyperbola, can be described with quadrics.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• quadratic polynomial — noun a polynomial of the second degree • Syn: ↑quadratic • Hypernyms: ↑polynomial, ↑multinomial …   Useful english dictionary

• Complex quadratic polynomial — A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers. Contents 1 Forms 2 Conjugation 2.1 Between forms 2.2 With doubling map …   Wikipedia

• Quadratic probing — is a scheme in computer programming for resolving collisions in hash tables.Quadratic probing operates by taking the original hash value and adding successive values of an arbitrary quadratic polynomial to the starting value. This algorithm is… …   Wikipedia

• Quadratic sieve — The quadratic sieve algorithm (QS) is a modern integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is… …   Wikipedia

• Quadratic — In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. Quadratus is Latin for square . Mathematics Algebra… …   Wikipedia

• Quadratic function — A quadratic function, in mathematics, is a polynomial function of the form f(x)=ax^2+bx+c ,!, where a e 0 ,!. The graph of a quadratic function is a parabola whose major axis is parallel to the y axis.The expression ax^2+bx+c in the definition of …   Wikipedia