Quadratic polynomial

Quadratic polynomial

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(x) = a_2 x^2 + a_1 x + a_0 ,.
* logistic form, f_r(x) = r x ( 1-x ) ,, used to study 1D discrete dynamics,
* monic and centered form, f_c(x) = 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_{i, j = 1}^{n} Q_{i,j} x_i x_j + sum_{i = 1}^{n} P_i x_i + Rwhere "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.

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