`(x^n-a^n)` is completely divisible by (x - a) is

Options

A. n is an even natural number
B. n is an odd natural number
C. n is a natural number
D. nisaprime

Answer

To determine whether \( (x^n - a^n) \) is completely divisible by \( (x - a) \), we can use the concept of polynomial factorization and the Remainder Theorem. Here’s a step-by-step solution:

### Step 1: Understand the expression
We start with the expression \( (x^n - a^n) \). We want to check if this expression is divisible by \( (x - a) \).

### Step 2: Apply the Remainder Theorem
According to the Remainder Theorem, if a polynomial \( f(x) \) is divided by \( (x - c) \), the remainder is \( f(c) \). In our case, let \( f(x) = x^n - a^n \) and \( c = a \).

### Step 3: Calculate \( f(a) \)
Now, we calculate \( f(a) \):
\[ f(a) = a^n - a^n = 0 \]
Since \( f(a) = 0 \), it implies that \( (x - a) \) is a factor of \( (x^n - a^n) \).

### Step 4: Conclusion
Thus, we conclude that \( (x^n - a^n) \) is completely divisible by \( (x - a) \) for any natural number \( n \).

### Final Answer
Yes, \( (x^n - a^n) \) is completely divisible by \( (x - a) \) for any natural number \( n \).

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