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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 ...
Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. [8] For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the ...
Degree: The maximum exponents among the monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials of the polynomial. Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p.
The minimal polynomial over K of θ is thus the monic polynomial of minimal degree that has θ as a root. Because L is a field, this minimal polynomial is necessarily irreducible over K. For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number i is +.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the n th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases.
The degree is the sum of the exponents on the variables; in this example, = + + A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example, + + is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.
Examples of graded algebras are common in mathematics: Polynomial rings. The homogeneous elements of degree n are exactly the homogeneous polynomials of degree n. The tensor algebra of a vector space V. The homogeneous elements of degree n are the tensors of order n, .
A solvable quintic is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals. To characterize solvable quintics, and more generally solvable polynomials of higher degree, Évariste Galois developed techniques which gave rise to group theory and Galois theory.