Sum of the coefficients of `(1+2x-4x^2)^(2003)`

Options

A. 1
B. 2
C. 3
D. `-1`

Answer

To find the sum of the coefficients of the expression \((1 + 2x - 4x^2)^{2003}\), we can follow these steps:

### Step 1: Understand the Problem
We need to find the sum of the coefficients of the polynomial when expanded. The sum of the coefficients of a polynomial \(P(x)\) can be found by evaluating \(P(1)\).

### Step 2: Substitute \(x = 1\)
We substitute \(x = 1\) into the expression:
\[ P(1) = (1 + 2(1) - 4(1)^2)^{2003} \]

### Step 3: Simplify the Expression
Now, simplify the expression inside the parentheses:
\[ P(1) = (1 + 2 - 4)^{2003} \]
\[ P(1) = (3 - 4)^{2003} \]
\[ P(1) = (-1)^{2003} \]

### Step 4: Evaluate the Power
Since \(2003\) is an odd number, we know that:
\[ (-1)^{2003} = -1 \]

### Conclusion
Thus, the sum of the coefficients of the polynomial \((1 + 2x - 4x^2)^{2003}\) is \(-1\).

### Final Answer
The sum of the coefficients is \(-1\).

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