Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer.  The remaining $a$'s are integers or zero.  Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$.  [See example 25 for the meaning of $|x|$.]  The number of polynomials with $h=3$ is:
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $
We need to find the number of polynomials of the form \(a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n\) such that \(h = n + a_0 + |a_1| + |a_2| + \cdots + |a_n| = 3\). Here, \(n\) is a non-negative integer, \(a_0\) is a positive integer, and the remaining \(a_i\) are integers or zero.

We will perform casework on \(n\):

1. **Case \(n = 3\):**
   \[
   h = 3 + a_0 + |a_1| + |a_2| + |a_3|
   \]
   Since \(h = 3\), we have:
   \[
   3 + a_0 + |a_1| + |a_2| + |a_3| = 3 \implies a_0 + |a_1| + |a_2| + |a_3| = 0
   \]
   Since \(a_0\) is a positive integer, this is impossible. Thus, there are no polynomials for \(n = 3\).

2. **Case \(n = 2\):**
   \[
   h = 2 + a_0 + |a_1| + |a_2|
   \]
   Since \(h = 3\), we have:
   \[
   2 + a_0 + |a_1| + |a_2| = 3 \implies a_0 + |a_1| + |a_2| = 1
   \]
   Since \(a_0\) is a positive integer, the only possibility is \(a_0 = 1\) and \(|a_1| = |a_2| = 0\). This gives the polynomial:
   \[
   x^2
   \]
   Thus, there is 1 polynomial for \(n = 2\).

3. **Case \(n = 1\):**
   \[
   h = 1 + a_0 + |a_1|
   \]
   Since \(h = 3\), we have:
   \[
   1 + a_0 + |a_1| = 3 \implies a_0 + |a_1| = 2
   \]
   - If \(a_0 = 1\), then \(|a_1| = 1\). This gives two polynomials:
     \[
     x + 1 \quad \text{and} \quad x - 1
     \]
   - If \(a_0 = 2\), then \(|a_1| = 0\). This gives one polynomial:
     \[
     2x
     \]
   Thus, there are 3 polynomials for \(n = 1\).

4. **Case \(n = 0\):**
   \[
   h = 0 + a_0
   \]
   Since \(h = 3\), we have:
   \[
   a_0 = 3
   \]
   This gives the polynomial:
   \[
   3
   \]
   Thus, there is 1 polynomial for \(n = 0\).

Summarizing all the cases, the set of polynomials is:
\[
\{x^2, x+1, x-1, 2x, 3\}
\]

The final answer is \(\boxed{5}\)