To find the roots of the equation \(x^9 + x^8 - x - 1 = 0\), we can use various methods like synthetic division, substitution, or numerical methods. Here’s a basic approach to check for simple roots:

### Try simple solutions

1. **Substitute \(x = 1\):**

\[
1^9 + 1^8 - 1 - 1 = 1 + 1 - 1 - 1 = 0
\]

So, \(x = 1\) is a root of the equation.

### Factor the equation

Since \(x = 1\) is a root, \(x - 1\) is a factor. We can perform polynomial division to find other factors.

### Use polynomial division

1. Divide \(x^9 + x^8 - x - 1\) by \(x - 1\) using synthetic or long division.

The division will leave a polynomial of degree 8. Solving this reduced polynomial may require numerical methods or graphing to find additional roots.

### Numerical or Graphing Solution

For equations of higher degrees with no obvious analytical solutions:

1. **Graphing:** Use software or graphing calculators to visualize and find the roots.
2. **Numerical Methods:** Use methods like Newton-Raphson or software tools for approximated solutions to the remaining roots.

So, to begin with, one known root is \(x = 1\), and further analysis or numerical approximation will be necessary for other potential roots.