- Quadratic polynomial
In mathematics, a

**quadratic polynomial**is apolynomial 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=Thecoefficients 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 aquadratic function in one variable. An example is given by "f"("x") = "x"^{2}+ "x" − 2;. The graph of such a function is aparabola (in degenerate cases a line), and its zeroes can be found by solving thequadratic 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 studycomplex dynamics .**The general case**In the general case, a quadratic polynomial in "n" variables "x"

_{1}, ..., "x"_{"n"}can be written in the form:$sum\_\{i,\; j\; =\; 1\}^\{n\}\; Q\_\{i,j\}\; x\_i\; x\_j\; +\; sum\_\{i\; =\; 1\}^\{n\}\; P\_i\; x\_i\; +\; R$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 aquadric . Theconic section s, such asellipse andhyperbola , can be described with quadrics.

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