1. **Understand the Polynomial and its Roots**: The polynomial given is \(x^2 + kx + 36\). Let the roots of the polynomial be \(r_1\) and \(r_2\). 2. **Relationships Between Roots and Coefficients**: For a polynomial \(x^2 + bx + c = 0\), the sum of the roots \(r_1 + r_2 = -b\) and the product of the roots \(r_1 \cdot r_2 = c\). In our case, \(r_1 + r_2 = -k\) and \(r_1 \cdot r_2 = 36\). 3. **Conditions for Distinct Integer Roots**: We need to find pairs of distinct integers \((r_1, r_2)\) such that their product is 36. Then, for each pair, we will determine the corresponding value of \(k\). 4. **Check for Distinct Pairs**: Ensure \(r_1 \neq r_2\). 5. **Calculate the Values of \(k\)**: For each valid pair \((r_1, r_2)\), compute \(k = -(r_1 + r_2)\) and count the distinct values. Let's implement this in Python to find the number of distinct values of \(k\). ```python import sympy as sp # Define the product of the roots product = 36 # Find all pairs of integers (r1, r2) such that r1 * r2 = product integer_pairs = [] for i in range(1, product + 1): if product % i == 0: j = product // i integer_pairs.append((i, j)) integer_pairs.append((j, i)) # Include both (i, j) and (j, i) # Filter pairs to ensure they are distinct and r1 != r2 distinct_pairs = [(r1, r2) for r1, r2 in integer_pairs if r1 != r2] # Calculate the corresponding values of k k_values = set(-(r1 + r2) for r1, r2 in distinct_pairs) # Output the number of