Find the values of k for which x$$^2$$​ + 5kx + k$$^2$$​ + 5 is exactly divisible by x + 2 but not divisible by x + 3.

Correct option is D

Given:

x² + 5kx + k² + 5 is exactly divisible by (x + 2), and not divisible by (x + 3).

Solution:

To find the value of k:

Use x = -2 in the polynomial

=> (-2)² + 5k × (-2) + k² + 5 = 0

=> 4 -10k + k² + 5 = 0

=> k² - 10k + 9 = 0

=> k² - 9k - k + 9 = 0

=> k(k - 9) - 1(k - 9) = 0

=> (k - 9)(k - 1) = 0

=> k = 9 or k = 1----(1)

Use x = -3 in the polynomial

=> (-3)² + 5k × (-3) + k² + 5 ≠ 0

=> 9 -15k + k² + 5 ≠ 0

=> k² - 15k + 14 ≠ 0

=> (k - 14)(k - 1) ≠ 0

=> k ≠ 14 and k ≠ 1----(2)

Conclusion:

From (1) and (2), only k = 9 satisfies both conditions.

=> Final Answer: k = 9