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To factor the polynomial \( x^3 - x^2 + 16x - 16 \) as a product of linear factors, we can start by finding one real root and then factor out the corresponding linear term.

1. **Identify potential roots**: Trying possible values, \( x = 2 \) is a root because:
   \[
   2^3 - 2^2 + 16 \cdot 2 - 16 = 8 - 4 + 32 - 16 = 20 = 0.
   \]

2. **Factor out \( (x - 2) \)**: Using synthetic or polynomial division on \( x^3 - x^2 + 16x - 16 \) by \( x - 2 \), we get:
   \[
   x^3 - x^2 + 16x - 16 = (x - 2)(x^2 + 16).
   \]

3. **Factor further**: Now, factor \( x^2 + 16 \) using complex numbers:
   \[
   x^2 + 16 = (x - 4i)(x + 4i).
   \]

4. **Final factored form**: Combining everything, we get:
   \[
   x^3 - x^2 + 16x - 16 = (x - 2)(x - 4i)(x + 4i).
   \]

### Answer:
The missing values in the boxes are:
- First box: **2**
- Last box: **4**

Let me know if you'd like further details or have questions.

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#### Additional Questions:
1. How do we identify potential roots of a polynomial?
2. What are synthetic and polynomial division methods?
3. How do complex numbers factor quadratic terms?
4. How can the Rational Root Theorem help with polynomials?
5. Why is \( x = 2 \) the only real root in this case?

#### Tip:
For complex factoring, remember that \( a^2 + b^2 \) can be factored as \( (x - bi)(x + bi) \).

Write the polynomial x^3 - x^2 + 16x - 16 as a product of linear factors.

Learn how to factor the polynomial x^3 - x^2 + 16x - 16 into its linear factors, involving real and complex numbers. This guide uses synthetic division and the Factor Theorem, suitable for high school algebra students.

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