If sum of the coefficients in the expansion `(2x^(2)-3cx+c^(2))^(17)` is zero , then c is equal to

Options

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

Answer

To solve the problem, we need to find the value of \( c \) such that the sum of the coefficients in the expansion of \( (2x^2 - 3cx + c^2)^{17} \) is equal to zero.

### Step-by-Step Solution:

1. **Understanding the Sum of Coefficients**:
The sum of the coefficients in a polynomial can be found by substituting \( x = 1 \) into the polynomial. This means we need to evaluate \( (2(1)^2 - 3c(1) + c^2)^{17} \).

2. **Substituting \( x = 1 \)**:
Substitute \( x = 1 \) into the polynomial:
\[ 2(1)^2 - 3c(1) + c^2 = 2 - 3c + c^2 \]
Therefore, the sum of the coefficients becomes:
\[ (2 - 3c + c^2)^{17} \]

3. **Setting the Sum of Coefficients to Zero**:
We want this expression to equal zero:
\[ (2 - 3c + c^2)^{17} = 0 \]
Since a power is zero only when the base is zero, we set the base equal to zero:
\[ 2 - 3c + c^2 = 0 \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ c^2 - 3c + 2 = 0 \]

5. **Factoring the Quadratic**:
We can factor the quadratic equation:
\[ (c - 1)(c - 2) = 0 \]

6. **Finding the Roots**:
Setting each factor to zero gives us the possible values for \( c \):
\[ c - 1 = 0 \quad \Rightarrow \quad c = 1 \]
\[ c - 2 = 0 \quad \Rightarrow \quad c = 2 \]

7. **Conclusion**:
The values of \( c \) that satisfy the condition are \( c = 1 \) and \( c = 2 \).

### Final Answer:
The values of \( c \) are \( c = 1 \) and \( c = 2 \).