1. ## subspace help

i dont understand subspaces at all
can you help me understand the question
in the capture below.

2. Do you understand vector spaces at all? Because if you do, subspaces should be easy. A subspace of a vector space is any subset that forms a space in its own right. And they are easier to identity because you already know that all the requirements on the operations, commutative, associative, etc. are automatically satisfied. You really only need to look at three things:
1) Does the set contain the 0 vector?
2) Is the set "closed" under additon- that is, if you add two things in the set is the sum also in the set?
3) Is the set "closed" under scalar multiplication- that is, if you multiply something in the set by a scalar (number) is the result also in the set?

Your vector space is $\displaystyle P^4$, the set of all polynomials over the real numbers of degree 4 or less.
The subsets are

a) The set of polynomials of degree 4.
What is the "0 vector" for $\displaystyle P^4$? Is it in this set?
Or, look at $\displaystyle p(x)= x^4- 3x^2+ 2x$ and $\displaystyle q= -x^4+ 2x- 3$, both polynomials of degree 4. Is their sum a polynomial of degree 4?
Or, if you multiply a polynomial of degree 4 by the number 0 is the result a polynomial of degree 4?

b) The set of polynomials of the form $\displaystyle (t^2+ 1)q(t)$ where q(t) is a polynomial of degree 2 or less (so that the product $\displaystyle (t^2+1)q(t)$ is a polynomial of degree 4 or less).
Is the 0 vector of this form? $\displaystyle 0= (t^2+1)q(t)$ for what q?
Is the sum of two polynomials of this form also of this form? What is $\displaystyle (t^2+1)p(t)+ (t^2+ 1)q(t)$ where p and q are polynomials of degree 2 or less?
Is a number times a polynomial of this form also of this form? What is $\displaystyle k(t^2+1)q(t)$? Can it be written as $\displaystyle (t^2+1)p(t)$ for some polynomial p? If so what is p(t) in terms of k and q(t)?

3. just to add,the zero polynomial has a degree $\displaystyle -\infty$.
$\displaystyle deg(\mathbf{0})=-\infty$
so it's obvious that the set of polynomials with degree 4 doesn't contain the zero polynomial.
in general :
Any set of polynomials with degree equal or less than a natural $\displaystyle n$,is a subspace of $\displaystyle \mathbb{K}[x]$.