Assertion (A) : The Remainder obtained when the polynomial `x^(64)+x^(27)+1` is divided by `x+1` is 1
Reason (R) : If `f(x)` is divided by `x-a` then the remainder is `f(a)`

Options

A. Both A & R are true and R is correct explanation of A
B. Both A & R are true and R is not correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem, we need to analyze the given assertion and reason using the Remainder Theorem.

### Step 1: Define the polynomial
Let \( f(x) = x^{64} + x^{27} + 1 \).

### Step 2: Apply the Remainder Theorem
According to the Remainder Theorem, when a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). In our case, we are dividing by \( x + 1 \), which can be rewritten as \( x - (-1) \). Therefore, we need to find \( f(-1) \).

### Step 3: Calculate \( f(-1) \)
Now, we will substitute \( -1 \) into the polynomial:
\[ f(-1) = (-1)^{64} + (-1)^{27} + 1 \]
Calculating each term:
- \( (-1)^{64} = 1 \) (since 64 is even)
- \( (-1)^{27} = -1 \) (since 27 is odd)

Now substituting these values back into the equation:
\[ f(-1) = 1 - 1 + 1 = 1 \]

### Step 4: Conclusion for the assertion
Since \( f(-1) = 1 \), the remainder when \( f(x) \) is divided by \( x + 1 \) is indeed 1. Therefore, the assertion \( A \) is true.

### Step 5: Analyze the reason
The reason states that if \( f(x) \) is divided by \( x - a \), then the remainder is \( f(a) \). This is a correct statement according to the Remainder Theorem. Therefore, the reason \( R \) is also true.

### Final Conclusion
Both the assertion \( A \) and the reason \( R \) are true, and \( R \) is a correct explanation of \( A \). Thus, the correct option is the first one: both \( A \) and \( R \) are true and \( R \) is a correct explanation of \( A \).

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