Quadratic polynomial
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In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two. A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z.
Any single-variable quadratic polynomial may be written as
where x is the variable, and a, b, and c represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation ax2 + bx + c = 0. The solutions to this equation are called the roots of the quadratic polynomial, and may be found through factorization or through the use of the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.
Any quadratic polynomial with two variables may be written as
where x and y are the variables and a, b, c, d, e, and f are the coefficients. Such polynomials are fundamental to the study of conic sections. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces. In linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space.
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[edit] 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.
[edit] 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.
[edit] Variables
[edit] 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) = x2 + 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 three main forms :
- general form,
. - logistic form,
, used to study 1D discrete dynamics, - monic and centered form,
, used to study complex dynamics.
[edit] The general case
In the general case, a quadratic polynomial in n variables x1, ..., xn can be written in the form
where Q is a symmetric n-dimensional matrix, P is an n-dimensional vector, and R a constant.
[edit] Roots
The zeroes of a quadratic polynomial form a quadric. The conic sections, such as ellipse and hyperbola, can be described with quadrics.
