Final answer:
The set of all fifth-degree polynomials is a vector space because it satisfies the required axioms such as closure under addition and scalar multiplication, the existence of a zero vector, additive inverses, and adherence to associative and distributive laws.
Step-by-step explanation:
The question pertains to whether the set of all fifth-degree polynomials forms a vector space. A vector space is defined as a collection of objects, known as vectors, that can be added together and multiplied by scalars while satisfying certain axioms. The set of all polynomials of a fixed degree n, such as fifth-degree polynomials, does indeed form a vector space over a field, generally the field of real numbers or complex numbers.
The axioms that need to be verified for the set of fifth-degree polynomials to be considered a vector space include closure under addition and scalar multiplication, the existence of a zero vector (a polynomial where all coefficients are zero), the existence of additive inverses for each vector, and adherence to associative and distributive laws. Fifth-degree polynomials meet these criteria because:
- Adding any two fifth-degree polynomials results in another fifth-degree polynomial.
- Multiplying any fifth-degree polynomial by a scalar results in another fifth-degree polynomial.
- The zero polynomial, which has all coefficients equal to zero, serves as the zero vector in this space.
- Each fifth-degree polynomial has an additive inverse, which is obtained by negating all of its coefficients.
- The set is closed under addition and scalar multiplication, and it satisfies the associative property for addition, the commutative property for addition, and the distributive properties.
Hence, the set of all fifth-degree polynomials is indeed a vector space.