Subspaces in Polynomial P_5(x) of Degree < 5

[gtfitzpatrick]

If P_{5}(x) is the set of all polynomials in x in degree less than 5. Which of following subsets of P_{5}(x) are subspaces.
(i) the set of all polynomials in P_{5}(x) of even degree
(ii) the set of all polynomials in P_{5}(x) of degree 3
(iii) the set of all polynomials p(x) in P_{5}(x) such that p(x)=0
(iv) the set of all polynomials p(x) in P_{5}(x) such that p(x)=0 has at least one real root

i'm really not sure but this is what i think
polynomial =ax^{4}+bx^{3}+cx^{2}+dx+e
(i)im not sure about the question but i think it means

a1x^{4}+b1x^{3}+c1x^{2}+d1x+e1
+
c2x^{2}+d2x+e2

which i think are the 2 even degree polynomials so i add them and see if the answer is also in P_{5}(x)
a1x^{4}+b1x^{3}+(c1+c2)x^{2}+(d1+d2)x+(e1+e2)
since the resulting polynomial is still of degree 4 it is in P_{5}(x) and so is a subspace?

(ii)by much the same reasoning

b1x^{3}+c1x^{2}+d1x+e1 is of degree 3 and so is not a subspace?

i don't understand questions (iii) & (iv), please pointers anyone then i'll try to do them

 

[Office_Shredder]

Definition of a subspace: It's a vector space, and a subset of your original vector space. The common way to test that something is a subspace is to see if it's closed under linear transformations.

So for (ii), a degree 3 polynomial is always of degree less than 5, so you have to test if it's closed under linear combinations. Similarly for (i), you didn't add two arbitrary even degree polynomials, and that can get you in trouble. For example, what about these two:

x⁴+x-1 and -x⁴? When you add them are you still in the subset of even degree polynomials? Try something similar for (ii).

For (iii), I'm not sure what they want, but it looks like they're asking if {0} is a subspace maybe

For (iv), this is the set of all p(x) such that there exists some r a real number with p(r)=0 (as opposed to, say, x²+1 which has no real roots). So you need to see if adding two polynomials which each has a real root gives another polynomial which has a real root