If (x - 2k) is a factor of f(x), which of the following must be true? 
f(2k) = 0
f(-2k) = 0
A root of f(x) is x = -2k.
Any intercept of f(x) is x = 2k.

Solution:

Factor theorem states that a polynomial f(x) has a factor (x - k) if and only if f(k) = 0 (i.e. k is a root).

If (x – 2k) is a factor of f(x), then by the factor theorem f(2k) = 0

f(2k) = 0, so option (1) is true.

f(2k) = 0 so option (2) is false.

x = 2k is a root, not x = -2k so option (3) is false.

The x intercept of f(x) is x = 2k not the y intercept, so option (4) is false.

Therefore, option (1) is true.

If (x - 2k) is a factor of f(x), which of the following must be true?

Summary:

If (x - 2k) is a factor of f(x), option (1) - f(2k) = 0 is true.