The equation whose roots are opposite in sign and equal to magnitude of the roots of ` x^4 -5x^3 +11x +3=0` is

Options

A. `x^4 +5x^3 -11 x-3=0`
B. `x^4 +5x^3 -11x +3=0`
C. `x^4 +5x^3-11x +6=0`
D. ` x^4 +5x^3 - 9x +3=0`

Answer

To solve the problem, we need to find the equation whose roots are opposite in sign and equal in magnitude to the roots of the polynomial \( x^4 - 5x^3 + 11x + 3 = 0 \).

### Step-by-Step Solution:

1. **Identify the Given Polynomial**:
We start with the polynomial:
\[ P(x) = x^4 - 5x^3 + 11x + 3 \]

2. **Understanding the Roots**:
The roots of \( P(x) \) are denoted as \( r_1, r_2, r_3, r_4 \). We need to find a new polynomial whose roots are \( -r_1, -r_2, -r_3, -r_4 \). This means that the new roots are the negatives of the original roots.

3. **Substituting \( -x \)**:
To find the polynomial with roots \( -r_i \), we substitute \( x \) with \( -x \) in the original polynomial:
\[ P(-x) = (-x)^4 - 5(-x)^3 + 11(-x) + 3 \]
Simplifying this, we get:
\[ P(-x) = x^4 + 5x^3 - 11x + 3 \]

4. **Formulating the New Polynomial**:
The new polynomial whose roots are \( -r_1, -r_2, -r_3, -r_4 \) is:
\[ x^4 + 5x^3 - 11x + 3 = 0 \]

5. **Identifying the Correct Option**:
Now we compare this polynomial with the given options:
- Option 1: \( x^4 + 5x^3 - 11x - 3 = 0 \)
- Option 2: \( x^4 + 5x^3 - 11x + 3 = 0 \)
- Option 3: \( x^4 + 5x^3 - 11x + 6 = 0 \)
- Option 4: \( x^4 + 5x^3 - 9x + 3 = 0 \)

The polynomial we derived matches **Option 2**:
\[ x^4 + 5x^3 - 11x + 3 = 0 \]

### Final Answer:
The equation whose roots are opposite in sign and equal in magnitude to the roots of \( x^4 - 5x^3 + 11x + 3 = 0 \) is:
\[ \boxed{x^4 + 5x^3 - 11x + 3 = 0} \]