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A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M n (R).
A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations. Two equations are equivalent if they have the same set of solutions.
Let () be a polynomial equation, where P is a univariate polynomial of degree n. If one divides all coefficients of P by its leading coefficient, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial. For example, the equation
Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.
Such polynomials often arise in a quadratic equation + + = The solutions to this equation are called the roots and can be expressed in terms of the coefficients as the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.
An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree: linear equation for degree one; quadratic equation for degree two; cubic equation for degree three; quartic equation for degree four; quintic equation for degree five
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points. + + + + = where a ≠ 0.