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To solve this problem, we will apply the **Conjugate Zeros Theorem**, which states that if a polynomial has real coefficients and a complex zero, then the conjugate of that complex zero is also a zero.

### Given Information
The polynomial is:
\[
P(x) = x^3 - 7x^2 + 25x - 39
\]
We are told that \( 2 + 3i \) is a zero of \( P(x) \).

### Step-by-Step Solution
1. **Use the Conjugate Zeros Theorem**: Since \( P(x) \) has real coefficients, the conjugate \( 2 - 3i \) must also be a zero.

2. **Determine the Remaining Zero**: Since this is a third-degree polynomial, it has exactly three zeros. We have already identified two zeros: \( 2 + 3i \) and \( 2 - 3i \). The remaining zero must be a real number.

3. **Find the Real Zero**: Since the polynomial can be factored as \((x - (2 + 3i))(x - (2 - 3i))(x - r)\) for some real \( r \), we can simplify \((x - (2 + 3i))(x - (2 - 3i))\) first:
   \[
   (x - (2 + 3i))(x - (2 - 3i)) = (x - 2 - 3i)(x - 2 + 3i)
   \]
   This expression expands as a difference of squares:
   \[
   = (x - 2)^2 - (3i)^2 = x^2 - 4x + 13
   \]
   So we can rewrite \( P(x) \) as:
   \[
   P(x) = (x^2 - 4x + 13)(x - r)
   \]
   To find \( r \), we can equate coefficients by expanding \((x^2 - 4x + 13)(x - r)\) and matching it with \( P(x) = x^3 - 7x^2 + 25x - 39 \). However, a quicker approach is to use **synthetic division** or **trial and error** to check if \( x = 3 \) is a zero of \( P(x) \).

4. **Verify \( x = 3 \) as a Zero**:
   \[
   P(3) = 3^3 - 7 \cdot 3^2 + 25 \cdot 3 - 39 = 27 - 63 + 75 - 39 = 0
   \]
   Therefore, \( x = 3 \) is indeed a zero.

### Solution Summary
The zeros of \( P(x) = x^3 - 7x^2 + 25x - 39 \) are:
\[
2 + 3i, \; 2 - 3i, \; 3
\]

Would you like more details on any step, or do you have questions?

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### Related Questions

1. How does the Conjugate Zeros Theorem apply to polynomials with real coefficients?
2. How do you perform synthetic division to confirm zeros of polynomials?
3. What are other methods for finding zeros of a polynomial if one zero is known?
4. How do you factor polynomials with complex roots?
5. Why does a third-degree polynomial have exactly three zeros?

### Tip
When dealing with complex roots in real polynomials, always consider the conjugate pair. This simplifies the factorization process.

Find all other zeros of P(x) = x^3 - 7x^2 + 25x - 39, given that 2 + 3i is a zero.

Explore a step-by-step solution to find all zeros of the polynomial P(x) = x^3 - 7x^2 + 25x - 39, given that 2 + 3i is a zero. This solution uses the Conjugate Zeros Theorem, polynomial factorization, and synthetic division to determine the complete set of roots. Suitable for students in Grades 10-12.

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