|
The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its canonical form consisting of a linear combination of monomials. The degree of a term is the sum of the exponents of the variables that appear in it. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial). For example, the polynomial has three terms. (Notice, this polynomial can also be expressed as .) The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5 which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form (for example ), one has to put it first in standard form by expanding the products (by distributivity) and combining the like terms; for example is of degree 1, even though each summand has degree 2. However, this is not needed when the polynomial is expressed as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors. ==Names of polynomials by degree== The following names are assigned to polynomials according to their degree:〔(【引用サイトリンク】 title=Names of Polynomials )〕〔Mac Lane and Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic". (p. 107)〕〔King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".〕 * Special case – zero (see § Degree of the zero polynomial below) * Degree 0 – constant〔Shafarevich (2003) says of a polynomial of degree zero, : "Such a polynomial is called a ''constant'' because if we substitute different values of ''x'' in it, we always obtain the same value ." (p. 23)〕 * Degree 1 – linear * Degree 2 – quadratic * Degree 3 – cubic * Degree 4 – quartic * Degree 5 – quintic * Degree 6 – sextic (or, less commonly, hexic) * Degree 7 – septic (or, less commonly, heptic) For higher degrees, names have sometimes been proposed,〔James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. ((''Mechanics Magazine'', Vol. LV, p. 171 ))〕 but they are rarely used: * Degree 8 – octic * Degree 9 – nonic * Degree 10 – decic Names for degree above three are based on Latin ordinal numbers, and end in ''-ic''. This should be distinguished from the names used for the number of variables, the arity, which are based on Latin distributive numbers, and end in ''-ary''. For example, a degree two polynomial in two variables, such as , is called a "binary quadratic": ''binary'' due to two variables, ''quadratic'' due to degree two. There are also names for the number of terms, which are also based on Latin distributive numbers, ending in ''-nomial''; the common ones are ''monomial'', ''binomial'', and (less commonly) ''trinomial''; thus is a "binary quadratic binomial". 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「degree of a polynomial」の詳細全文を読む スポンサード リンク
|