Problem Statement:
Find the value of K if X - 1 is a factor of P(X) = 2X^3 - 20X^2 + KX - 120.

Solution:
To find the value of K, we need to determine whether X - 1 is a factor of the given polynomial P(X). This can be done by applying the Remainder Factor Theorem.

Remainder Factor Theorem:
If a polynomial P(X) is divided by X - a, the remainder is P(a).

Step 1: Determine the Remainder
To apply the Remainder Factor Theorem, we substitute X = 1 into the polynomial P(X) and determine the remainder.

P(1) = 2(1)^3 - 20(1)^2 + K(1) - 120
= 2 - 20 + K - 120
= -138 + K

Therefore, the remainder when P(X) is divided by X - 1 is -138 + K.

Step 2: Determine the Value of K
Since X - 1 is a factor of P(X), the remainder should be equal to zero.

-138 + K = 0
K = 138

Therefore, the value of K is 138.

Explanation:
- We are given a polynomial P(X) = 2X^3 - 20X^2 + KX - 120.
- To determine whether X - 1 is a factor of P(X), we apply the Remainder Factor Theorem.
- By substituting X = 1 into P(X), we find the remainder to be -138 + K.
- Since X - 1 is a factor, the remainder should be zero. Therefore, -138 + K = 0.
- Solving for K, we find that K = 138.
- Hence, the value of K is 138.

Summary:
The value of K is 138.