Linear Algebra - Dimensions of Subspace How do I go about finding the dimension of the subspace:
$$$$S:={p($x$) ∈ $P_4$: p($x$)= 2p($x$) for  all $x\in\mathbb{R}$} of $P_4$

My textbook says $dim(P_n)=n+1$, but this does not give me the correct answer. All help is appreciated.
Let $n=4$, then $p \in S \leq P_n$ has the form of $\sum_{j=0}^{n}a_jx^j$ for some $a_j \in R$.  Therefore $p = 2p$ implies that for all $x \in R$,
$$\sum_{j=0}^{n}a_jx^j = 2\sum_{j=0}^{n}a_jx^j \iff 0 = \sum_{j=0}^{n}(2-1)a_jx^j = p(x).$$
What's the dimension of a space where every element has this property? 

Now that we are given $R = \mathbb R$, it's even easier: which element of $P_n \leq \mathbb R[x]$ has infinitely many roots?
I'm extremely confused by the equivalence drawn in linear algebra between vectors in Rn and polynomials. Can someone please point out the flaws in my understanding?

For Rn, a vector (1, 2, 3) has components in three different dimensions (along three different axes), so adding these together is impossible, right?

And when we express a system of linear equations in matrix form, say

2x + 3y + 10z = 45

we treat the variables x, y, z as living in 3 different dimensions that cannot be directly added together, just like the different dimensions ("axes") of the vector (1, 2, 3) in Rn, right?

My understanding (please correct me if I'm wrong) is that x y z somehow correspond to DIFFERENT dimensions, just as 1, 2, 3 lie on three different axes, so you can't just add them together like (1,2,3) = 1+2+3 = 6, nor can you do x+y+z = 3x, for example.

But in linear algebra I see people treat the same variable x* raised to different degrees, as different variables, exactly as in the previous case. So for

2x^3 + 3x^2 + 10x = 45

you would make a matrix with three columns, corresponding to the x^3, x^2, and x, exactly as if it had been

2x + 3y + 10z = 45

This feels totally wrong to me, because x^3 and x^2 all come from x!

I assume that since you can't just add variables of a different power together in a linear way, that's why they can be slotted into the matrix in this way? 

i.e.)

2x^2 + 3x^2 can be comined into 5x^2 but 2x^2 + 3x^3 cannot be combined.

But it still feels like a stretch to me and is completely baffling conceptually. How can you say that the same variable raised to different degrees lives in different dimensions, in the same way as the components of a vector in Rn??? For one, x^2 can be calculated if you know x, but a value living in dimension 2 of Rn, can never be calculated by just knowing a value in dimension 1 - they have totally different meanings.

If you graphed y = x and y = x^2, the two different graphs would look kind of different, but still be in the same dimension on the graph paper, right? One wouldn't be 3D or sticking out of the paper, right? So how do you draw this equivalency, that the 3 dimensional vector (1,2,3) corresponds to (x, x^2, x^3)?