If (x – 5) is a factor of f(x), which of the following must be true?
A) x = –5 and x = 5 are roots of f(x)
B) Neither x = –5 nor x = 5 is a root of f(x).
C) f(–5) = 0
D) f(5) = 0
On dividing two numbers, if we are getting remainder equal to zero, then the smaller number will be the factor of the larger number.
Answer: If (x – 5) is a factor of f(x), then f(5) = 0. Hence, Option D, f(5) = 0 is correct.
Let us proceed step by step to determine the correct option.
When we divide any polynomial by (x - a), we obtain a result of the form:
f (x) = (x - a) q (x) + f (a) [From Euclid's division algorithm]
If the remainder f (a) = 0, then (x − a) is a factor of f (x). --------(1)
Hence, from the given data,
(x - 5) is a factor of f(x).
Therefore, f (5) = 0 on substituting a = 5 [from equation (1)].
Thus, option D, f(5) = 0 is correct.