V not vector space with degree 3 polynomials

[chantella28]

Okay, so i have this problem in my text, and I've almost figured it out (i think) but i need a little help

"Let V be the set of all polynomials of degree 3. Define addition and scalar multiplication pointwise. Prove that V with respect to these operations of addiont and scalar multiplication is NOT a vector space"

I have pointwise addition and pointwise scalar multiplication defined... but I'm a little stumped on why V wouldn't be a vector space

-i know that V cannot have a 0 element (0 is not a degree 3 polynomial) and thus it cannot be a vector space, but that doesn't really explain it via addition/multiplication
-also, i know that the addition of 2 degree 3 polynomials does not always result in a degree 3 polynomial ie: x^3 + (-x^3)=0 (not a degree 3 polynomial)

if anyone has any other reasons why V cannot be a vector space according to pointwise addition/scalar multiplication please let me know

 

[EnumaElish]

Are you sure that 0 is not a 3rd degree polynomial?

 

[chantella28]

i don't know! :S

 

[amcavoy]

Your reasoning looks OK. You are talking about polynomials of only degree 3 right? Not P₃ which would represent all polynomials of degree 3 or less.

 

[HallsofIvy]

The (non) existence of the "0" vector certainly is a property of addition!

Or you look at (x³+ x)+ (x- x³). What does that tell you about closure of addition?

Or you could look at the scalar multiplication 0(x³).

 

[EnumaElish]

I thought that 0 = 0x³ is a 3rd degree polynomial, was I wrong?

 

[Muzza]

EnumaElish, that would mean that the polynomial x would also be of degree 3, since x = 0x^3 + x. Clearly this is not what we want. Usually, one defines the degree of a polynomial as the largest exponent n of x (or whatever the variable is called) such that the coefficient in front of x^n is not 0. (Sometimes one takes the degree of the zero polyonomial to be -1 or -infinity.)

 

[EnumaElish]

Hm, this makes sense, thanks.