Is the axiom for addition in the set of polynomials of degree >=3 true?

[stunner5000pt]

Determine if this is a vector space with the indicated operations

the set of V of all polynominals of degree >=3, togehter iwth 0, operations of P (P the set of polynomials)

now all the scalar multiplication axioms hold.
the text however says that the axion
[tex] \mbox{For u,v} \in V, \mbox{then} \ u+v \in V [/tex] does not hold

well ok take two polynomials
[tex] u(x) = a_{3} x^3 + ... + a_{n} x^n [/tex]
[tex] v(x) = b_{3} x^3 + ... + b_{k} x^k [/tex]
where both n,k>= 3, then suppose k< n
[tex] u(x) + v(x) = (a_{3} + b_{3}) x^3 + ... + (a_{k} + b_{k}) x^k + ... + a_{n} x^n [/tex]
which is certainly a polynomial or degree >= 3 isn't it?
It also applies for n<k and n = k
is the textbook wrong?

 

[matt grime]

What is the degree of the following poly

x^4+1

now, do you see your error?

 

[stunner5000pt]

ok let me correct that then
n,k >= 3
[tex] u(x) = a_{0} + a_{1} x + ... + a_{n} x^n [/tex]
[tex] v(x) = b_{0} + b_{1} x + ... + b_{k} x^k [/tex]
then for n< k
[tex] u + v = (a_{0} + b_{0}) + ... + (a_{k} + b_{k}) x^k + ... + a_{n} x^n [/tex]

stil lseems to be of degree three to me
however if k=n and an= -bn then the polynomial is no more degree 3
is this corret?

 

[matt grime]

Why don't you just find a counter example? two polys of degree 3 or greaterwhose sum isn't? A single counter examplem suffices.