Degree of a polynomial

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's 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. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.^[1] 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 ${\displaystyle 7x^{2}y^{3}+4x-9,}$ which can also be written as ${\displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},}$ 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, such as ${\displaystyle (x+1)^{2}-(x-1)^{2}}$, one can put it in standard form by expanding the products (by distributivity) and combining the like terms; for example, ${\displaystyle (x+1)^{2}-(x-1)^{2}=4x}$ is of degree 1, even though each summand has degree 2. However, this is not needed when the polynomial is written as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.