Degree of a polynomial

The degree of a polynomial is the highest of the degrees of its monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see order of a polynomial (disambiguation)).For example, the polynomial 7 x 2 y 3 + 4 x − 9 , {displaystyle 7x^{2}y^{3}+4x-9,} which can also be expressed as 7 x 2 y 3 + 4 x 1 y 0 − 9 x 0 y 0 , {displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},} has three terms. 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. The degree of a polynomial is the highest of the degrees of its monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see order of a polynomial (disambiguation)).For example, the polynomial 7 x 2 y 3 + 4 x − 9 , {displaystyle 7x^{2}y^{3}+4x-9,} which can also be expressed as 7 x 2 y 3 + 4 x 1 y 0 − 9 x 0 y 0 , {displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},} has three terms. 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: ( x + 1 ) 2 − ( x − 1 ) 2 {displaystyle (x+1)^{2}-(x-1)^{2}} ), one has to put it first in standard form by expanding the products (by distributivity) and combining the like terms; for example ( x + 1 ) 2 − ( x − 1 ) 2 = 4 x {displaystyle (x+1)^{2}-(x-1)^{2}=4x} 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. The following names are assigned to polynomials according to their degree: For higher degrees, names have sometimes been proposed, but they are rarely used: 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 x 2 + x y + y 2 {displaystyle x^{2}+xy+y^{2}} , 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 x 2 + y 2 {displaystyle x^{2}+y^{2}} is a 'binary quadratic binomial'.