Degree of A Polynomial - Behavior Under Addition, Subtraction, Multiplication and Function Composition

Behavior Under Addition, Subtraction, Multiplication and Function Composition

The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees, i.e.

.

.

E.g.

• The degree of is 3. Note that 3 ≤ max(3, 2)
• The degree of is 2. Note that 2 ≤ max(3, 3)

The degree of the product of a polynomial by a non-zero scalar is equal to the degree of the polynomial, i.e.

.

E.g.

• The degree of is 2, just as the degree of .

Note that for polynomials over a ring containing divisors of zero, this is not necessarily true. For example, in, but .

The collection of polynomials with coefficients from a given field F and degree smaller than or equal to a given number n thus forms a vector space. (Note, however, that this collection is not a ring, as it is not closed under multiplication, as is seen below.)

The degree of the product of two polynomials over a field is the sum of their degrees

.

E.g.

• The degree of is 3 + 2 = 5.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in, but .

The degree of the composition of two polynomials over a field or integral domain is the product of their degrees

.

E.g.

• If, then, which has degree 6.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in, but .