If 2,5,7,-4 are the roots of ` ax^4 + bx^3 + cx^2 + dx +e=0` then the roots of `ax^4 - bx^3 + cx^2 -dx +e=0` are

Options

A. `2,5,7,-4`
B. `-2,-5,-7,4`
C. `2,5,7,4`
D. `2,-5,7,-4`

Answer

To solve the problem, we need to find the roots of the equation \( ax^4 - bx^3 + cx^2 - dx + e = 0 \) given that the roots of the equation \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) are \( 2, 5, 7, -4 \).

### Step-by-Step Solution:

1. **Identify the Given Roots**:
The roots of the first equation \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) are given as \( 2, 5, 7, -4 \).

2. **Use the Transformation**:
To find the roots of the second equation \( ax^4 - bx^3 + cx^2 - dx + e = 0 \), we can use the substitution \( x = -x' \). This means we will replace \( x \) with \( -x \) in the original equation.

3. **Substitute \( x = -x' \)**:
When we substitute \( x = -x' \) into the first equation, we have:
\[ a(-x')^4 + b(-x')^3 + c(-x')^2 + d(-x') + e = 0 \]
Simplifying this gives:
\[ ax'^4 - bx'^3 + cx'^2 - dx' + e = 0 \]
This shows that the roots of the transformed equation are the negative of the roots of the original equation.

4. **Find the New Roots**:
Since the original roots are \( 2, 5, 7, -4 \), the new roots will be:
- For \( 2 \), the new root is \( -2 \)
- For \( 5 \), the new root is \( -5 \)
- For \( 7 \), the new root is \( -7 \)
- For \( -4 \), the new root is \( 4 \)

Therefore, the roots of the equation \( ax^4 - bx^3 + cx^2 - dx + e = 0 \) are \( -2, -5, -7, 4 \).

5. **Conclusion**:
The roots of the equation \( ax^4 - bx^3 + cx^2 - dx + e = 0 \) are \( -2, -5, -7, 4 \).

### Final Answer:
The roots are \( -2, -5, -7, 4 \).

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