Problem Statement:
Find the value of K, if (X + 5) is a factor of P(X) = -X^3 + X^3k?

Solution:
To find the value of K, we need to determine the relationship between the given polynomial P(X) and the factor (X + 5). This can be done by applying the factor theorem.

Factor Theorem:
If (X - a) is a factor of a polynomial P(X), then P(a) = 0.

Step 1: Applying the Factor Theorem
We need to check whether (X + 5) is a factor of P(X) = -X^3 + X^3k. For this, we substitute -5 for X in P(X) and check if the result is zero.

P(-5) = -(-5)^3 + (-5)^3k

Simplifying, we get:
P(-5) = -(-125) + (-125)k
P(-5) = 125 + (-125)k

Step 2: Checking for Zero Result
To establish that (X + 5) is a factor of P(X), the result of P(-5) should be zero.

125 + (-125)k = 0
125 = 125k
k = 1

Step 3: Conclusion
Hence, the value of K is 1, which makes (X + 5) a factor of P(X) = -X^3 + X^3k.

Summary:
- By applying the factor theorem, we found that the value of K is 1, which makes (X + 5) a factor of P(X) = -X^3 + X^3k.
- The factor theorem states that if a polynomial P(X) is divisible by (X - a), then P(a) = 0.
- In this case, we substituted -5 for X in P(X) and simplified to check if the result is zero.
- By solving the equation, we found that K = 1 satisfies the condition, and hence, (X + 5) is a factor of P(X).