# 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.

**Explanation:**

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]* *

where q (x) is a polynomial with one degree less than the degree of f(x), and f(a) is the remainder. This is called the remainder theorem.

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.

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