多項式の次数
↑ King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".
↑ 例えば以下のような用例がある: Shafarevich (2003) says of the zero polynomial: "In this case, we consider that the degree of the polynomial is undefined." (p. 27) Childs (1995) uses −1. (p. 233) Childs (2009) uses −∞ (p. 287), however he excludes zero polynomials in his Proposition 1 (p. 288) and then explains that the proposition holds for zero polynomials "with the reasonable assumption that − ∞ {\displaystyle -\infty } + m = − ∞ {\displaystyle -\infty } for m any integer or m = − ∞ {\displaystyle -\infty } ". Axler (1997) uses −∞.
二つの多項式の和(これには差も含めた意味で言う)の次数は、それらの多項式の次数のうち大きい方を超えない。式で書けば
deg  ( P ± Q ) ≤ max ( deg  ( P ) , deg  ( Q ) ) {\displaystyle \deg(P\pm Q)\leq \max(\deg(P),\deg(Q))}
が成り立つ。例えば
(x + x) + (x + 1) = x + x + x + 1 の次数は 3 で 3 ≤ max(3, 2) が成り立っている。
(x + x) − (x + x) = −x + x の次数は 2 で 2 ≤ max(3, 3) が成り立っている。