You're right. Let's clarify the distinct roots part. Here's the corrected analysis:

1. Distinct roots identification:

   • For the equation $x^2 - 1 = 0$: $x^2 - 1 = (x - 1)(x + 1)$ The distinct roots are $x = 1$ and $x = -1$.

   • For the equation $x^3 - 4x^2 + 4x = 0$: $x^3 - 4x^2 + 4x = x(x^2 - 4x + 4) = x(x - 2)^2$ The distinct roots are $x = 0$ and $x = 2$.

   • For the equation $x - 4 = 0$: $x - 4 = 0$ The distinct root is $x = 4$.

   • For the equation $x^2 - 6x + 9 = 0$: $x^2 - 6x + 9 = (x - 3)^2$ The distinct root is $x = 3$.

2. Listing all distinct roots:

   • $x = 1$
   • $x = -1$
   • $x = 0$
   • $x = 2$
   • $x = 4$
   • $x = 3$

3. Counting distinct roots:

   • There are 6 distinct roots: $1, -1, 0, 2, 4, 3$.

Therefore, the number of distinct roots is indeed 6. Your initial solution had the right idea but needed this clarification.

Please let me know if you have any questions or need further details.

Here are 5 questions you might ask next:

1. Can you show how to graph these polynomials and identify their roots visually?
2. What are some techniques for solving higher-degree polynomials?
3. How does the Fundamental Theorem of Algebra relate to the number of roots?
4. Can you explain the concept of root multiplicity in more detail?
5. How do complex roots affect the factorization of polynomials?

Tip: When solving polynomial equations, distinguishing between distinct roots and their multiplicities is crucial for understanding the polynomial's behavior and its graph.