The graph of the polynomial \[P(x) \equal{} x^5 \plus{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx \plus{} e\] has five distinct $ x$-intercepts, one of which is at $ (0,0)$. Which of the following coefficients cannot be zero?

$ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$
1. Given the polynomial \( P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e \) with five distinct \( x \)-intercepts, one of which is at \( (0,0) \), we know that \( P(0) = 0 \). This implies that \( e = 0 \).

2. Since \( P(x) \) has five distinct \( x \)-intercepts, we can write \( P(x) \) in its factored form as:
   \[
   P(x) = x(x - r_1)(x - r_2)(x - r_3)(x - r_4)
   \]
   where \( r_1, r_2, r_3, \) and \( r_4 \) are the non-zero roots of the polynomial.

3. Expanding the factored form, we get:
   \[
   P(x) = x(x - r_1)(x - r_2)(x - r_3)(x - r_4) = x^5 - (r_1 + r_2 + r_3 + r_4)x^4 + (r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4)x^3 - (r_1r_2r_3 + r_1r_2r_4 + r_1r_3r_4 + r_2r_3r_4)x^2 + r_1r_2r_3r_4x
   \]

4. Comparing coefficients with the original polynomial \( P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e \), we get:
   \[
   a = -(r_1 + r_2 + r_3 + r_4)
   \]
   \[
   b = r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4
   \]
   \[
   c = -(r_1r_2r_3 + r_1r_2r_4 + r_1r_3r_4 + r_2r_3r_4)
   \]
   \[
   d = r_1r_2r_3r_4
   \]
   \[
   e = 0
   \]

5. Since \( e = 0 \), we need to determine which of the coefficients \( a, b, c, \) or \( d \) cannot be zero. 

6. Notice that \( d = r_1r_2r_3r_4 \). Since \( r_1, r_2, r_3, \) and \( r_4 \) are distinct non-zero roots, their product \( r_1r_2r_3r_4 \) cannot be zero. Therefore, \( d \neq 0 \).

7. The coefficients \( a, b, \) and \( c \) can potentially be zero depending on the specific values of the roots \( r_1, r_2, r_3, \) and \( r_4 \). For example, if the sum of the roots \( r_1 + r_2 + r_3 + r_4 = 0 \), then \( a = 0 \). Similarly, specific combinations of the roots can make \( b \) or \( c \) zero.

8. Therefore, the coefficient that cannot be zero is \( d \).

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