Vector Space: Fifth-Degree Polynomials

AI Thread Summary
The discussion centers on determining whether the set of all fifth-degree polynomials forms a vector space under standard operations. It identifies that axioms 1, 4, 5, and 6 fail, leading to the conclusion that this set does not satisfy the requirements of a vector space. The failure of axiom 4 relates to the additive identity, which is not a fifth-degree polynomial, as it is a constant polynomial of degree zero. Axiom 5 fails because the additive inverse of a fifth-degree polynomial is not guaranteed to remain a fifth-degree polynomial. Overall, the set of fifth-degree polynomials does not meet the criteria to be classified as a vector space.
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15. Determine wheter the set is a vector space.
The set of all fifth-degree polynomials with the standard operations.
AXIOMS
1.u+v is in V
2.u+v=v+u
3.u+(v+w)=(u+v)+w
4.u+0=u
5.u+(-u)=0
6. cu is in V
7.c(u+v)=cu+cv
8.(c+d)u=cu+cd
9.c(du)=(cd)u
10.1(u)=u

the axioms that fail are 1,4,5, and 6. I don't know why 4,5,and 6 fail. Can anyone help me?
 
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How is the degree of a polynomial defined? Now, what is the additive identity, and is it a fifth-degree polynomial?
 

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