Let's go through your solution step by step to ensure it's correct:

1. Finding distinct roots:

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

   • For the equation $x - 4 = 0$: $x - 4 = 0$ The root is $x = 4$ with multiplicity 1.

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

2. Summarizing the roots and their multiplicities:

   • $x = 1$ (multiplicity 1)
   • $x = -1$ (multiplicity 1)
   • $x = 0$ (multiplicity 1)
   • $x = 2$ (multiplicity 2)
   • $x = 4$ (multiplicity 1)
   • $x = 3$ (multiplicity 2)

3. Counting the roots with multiplicities: The total number of roots (including multiplicities) is: $1 + 1 + 1 + 2 + 1 + 2 = 8$

4. Degree check: The degrees of the original polynomials are:

   • $x^2 - 1$: degree 2
   • $x^3 - 4x^2 + 4x$: degree 3
   • $x - 4$: degree 1
   • $x^2 - 6x + 9$: degree 2

   Adding the degrees: $2 + 3 + 1 + 2 = 8$

Thus, your calculation of the sum of degrees (8) is correct. However, your final degree calculation in the question states the total as 9, which is incorrect. The correct sum of degrees is 8.

Your conclusion about $n-1$ being $8-1 = 7$ should be $n-1 = 8-1 = 7$, not 9-1. Therefore, the final correct answer for $n-1$ is 7.

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

Here are 5 questions you might ask next:

1. Can you explain how to find the multiplicity of a root?
2. What are some common methods to factor polynomials?
3. How do you determine the degree of a polynomial?
4. Can you help with solving a polynomial equation using synthetic division?
5. How do the roots of a polynomial relate to its graph?

Tip: Always double-check your calculations and ensure the degrees of the polynomials and the number of roots match the expected total degrees and roots.