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The problem provided is as follows:

Given that \( x^2 - px + 1 \) is a factor of \( x^3 - 2x - 3r \), where \( p, r \in \mathbb{R} \) (real numbers) and \( p \leq 0 \), we are to find the values of \( p \) and \( r \).

### Approach

We know that if \( x^2 - px + 1 \) is a factor of \( x^3 - 2x - 3r \), then we can express:

\[
x^3 - 2x - 3r = (x^2 - px + 1)(x + a)
\]

where \( a \) is a constant that we need to determine. By expanding the right-hand side and comparing it with the given cubic expression, we will be able to find \( p \) and \( r \).

### Step 1: Expand \( (x^2 - px + 1)(x + a) \)

\[
(x^2 - px + 1)(x + a) = x^3 + ax^2 - px^2 - apx + x + a
\]
\[
= x^3 + (a - p)x^2 + (-ap + 1)x + a
\]

### Step 2: Compare with \( x^3 - 2x - 3r \)

Now, comparing the expanded form with \( x^3 - 2x - 3r \):

\[
x^3 + (a - p)x^2 + (-ap + 1)x + a = x^3 - 2x - 3r
\]

From this comparison, we obtain the following system of equations:

1. \( a - p = 0 \) (coefficient of \( x^2 \))
2. \( -ap + 1 = -2 \) (coefficient of \( x \))
3. \( a = -3r \) (constant term)

### Step 3: Solve the system of equations

From equation (1): 
\[
a = p
\]

Substitute \( a = p \) into equation (2):
\[
-p^2 + 1 = -2
\]
\[
-p^2 = -3
\]
\[
p^2 = 3
\]
\[
p = -\sqrt{3} \quad \text{(since \( p \leq 0 \))}
\]

Now substitute \( p = -\sqrt{3} \) into equation (3):
\[
p = -3r
\]
\[
-\sqrt{3} = -3r
\]
\[
r = \frac{\sqrt{3}}{3}
\]

### Final Answer:

- \( p = -\sqrt{3} \)
- \( r = \frac{\sqrt{3}}{3} \)

Would you like any further details or explanations? Here are some related questions:

1. How do we apply the factor theorem to find polynomial factors?
2. Why is \( p \leq 0 \) an important condition for determining \( p \)?
3. Can you explain how to expand polynomial expressions systematically?
4. How can we check if the calculated values of \( p \) and \( r \) are correct?
5. What is the significance of comparing coefficients when solving polynomial equations?

**Tip**: When solving problems involving polynomial division or factorization, always expand the product carefully and compare corresponding coefficients to form a system of equations.

Given that x^2 - px + 1 is a factor of x^3 - 2x - 3r, where p, r are real numbers and p ≤ 0, find the values of r and p.

Learn how to determine the values of r and p when given that x^2 - px + 1 is a factor of x^3 - 2x - 3r. This problem involves polynomial factorization, equation comparison, and solving systems of equations, suitable for students in Grades 10-12.

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