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 .
Read more about this topic: Degree Of A Polynomial
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