The remainder obtained when the polynominal `1+x+x^(3)+x^(9)+x^(27)+x^(81)+x^(243)` is divided by x-1 is

Options

A. 3
B. 5
C. 7
D. 11

Answer

To find the remainder when the polynomial \( P(x) = 1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243} \) is divided by \( x - 1 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of a polynomial \( P(x) \) when divided by \( x - c \) is simply \( P(c) \).

### Step-by-Step Solution:

**Step 1: Identify the polynomial and the divisor.**
- The polynomial is \( P(x) = 1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243} \).
- The divisor is \( x - 1 \).

**Step 2: Apply the Remainder Theorem.**
- According to the Remainder Theorem, the remainder when \( P(x) \) is divided by \( x - 1 \) is \( P(1) \).

**Step 3: Calculate \( P(1) \).**
- Substitute \( x = 1 \) into the polynomial:
\[ P(1) = 1 + 1 + 1^3 + 1^9 + 1^{27} + 1^{81} + 1^{243} \]

**Step 4: Simplify the expression.**
- Since \( 1^n = 1 \) for any integer \( n \):
\[ P(1) = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7 \]

**Step 5: State the remainder.**
- Therefore, the remainder when \( P(x) \) is divided by \( x - 1 \) is \( 7 \).

### Final Answer:
The remainder obtained when the polynomial \( 1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243} \) is divided by \( x - 1 \) is \( 7 \).

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