The remainder obtained when the polynomial `x^(4)-3x^(3)+9x^(2)-27x+81` is divided by x-3 is

Options

A. 81
B. 243
C. 405
D. 18

Answer

To find the remainder when the polynomial \( P(x) = x^4 - 3x^3 + 9x^2 - 27x + 81 \) is divided by \( x - 3 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( P(x) \) by \( x - c \) is simply \( P(c) \).

### Step-by-step solution:

1. **Identify the polynomial and the value of c**:
The polynomial is \( P(x) = x^4 - 3x^3 + 9x^2 - 27x + 81 \) and we are dividing by \( x - 3 \), so \( c = 3 \).

2. **Substitute c into the polynomial**:
We need to calculate \( P(3) \):
\[ P(3) = (3)^4 - 3(3)^3 + 9(3)^2 - 27(3) + 81 \]

3. **Calculate each term**:
- \( (3)^4 = 81 \)
- \( -3(3)^3 = -3 \times 27 = -81 \)
- \( 9(3)^2 = 9 \times 9 = 81 \)
- \( -27(3) = -81 \)
- \( +81 = 81 \)

4. **Combine the results**:
Now, we will add all the calculated terms together:
\[ P(3) = 81 - 81 + 81 - 81 + 81 \]
Simplifying this:
\[ P(3) = 81 - 81 + 81 - 81 + 81 = 81 \]

5. **Conclusion**:
The remainder when the polynomial \( P(x) \) is divided by \( x - 3 \) is \( 81 \).

### Final Answer:
The remainder obtained when the polynomial \( x^4 - 3x^3 + 9x^2 - 27x + 81 \) is divided by \( x - 3 \) is \( 81 \).

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