Subspace of P4: Polynomials of Even Degree

[amolv06]

Homework Statement

Determine whether the following is a subspace of $$P_{4}_$$

(a) The set of polynomials in $$P_{4}$$ of even degree.

Homework Equations

$$P_{4} = ax^{3}+bx^{2}+cx+d$$

The Attempt at a Solution

$$(p+q(x)) = p(x) + q(x)$$
$$(\alpha p)(x) = p(\alpha x)$$

If p and q are both of degree 2 then both scalar multiplication and vector addition should return a polynomial of degree 2 as far as I can tell, however my book states that this is not a subspace of $$P_{4}$$. I can't tell why it wouldn't be.

 

[HallsofIvy]

The set of "polynomials of degree 2" are specifically those of the form ax²+ bx+ c with a non-zero. In particular, that does not include the 0 vector. Further, if p₁(x)= x²+ 2x+ 1 and p₂(x)= -ax²+ 3x+ 1, thenthe sum is NOT a "polynomial of degree 2". The polynomials in P₄ that are of even degree are either of degree 2 or degree 0. In any case, the example I just gave answers your question.