Polynomial degree Definition and 7 Discussions

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. 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 written 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, such as



(
x
+
1

)

2


−
(
x
−
1

)

2




{\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,



(
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 written as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.

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