Find the transformed equation whose roots are the negatives of the roots of the equation
` x^4 +5x^3 +11 x+3=0`

Answer

To find the transformed equation whose roots are the negatives of the roots of the equation \( x^4 + 5x^3 + 11x + 3 = 0 \), we can follow these steps:

### Step 1: Define the original equation
Let the original polynomial be defined as:
\[ f(x) = x^4 + 5x^3 + 11x + 3 \]

### Step 2: Substitute \( x \) with \( -x \)
To find the transformed equation, we substitute \( x \) with \( -x \) in the polynomial \( f(x) \):
\[ f(-x) = (-x)^4 + 5(-x)^3 + 11(-x) + 3 \]

### Step 3: Simplify the expression
Now, we simplify the expression:
- The term \( (-x)^4 \) becomes \( x^4 \).
- The term \( 5(-x)^3 \) becomes \( -5x^3 \).
- The term \( 11(-x) \) becomes \( -11x \).
- The constant term \( 3 \) remains unchanged.

Putting it all together:
\[ f(-x) = x^4 - 5x^3 - 11x + 3 \]

### Step 4: Set the transformed equation to zero
The transformed equation, which has roots that are the negatives of the roots of the original equation, is:
\[ x^4 - 5x^3 - 11x + 3 = 0 \]

### Final Answer
Thus, the transformed equation is:
\[ x^4 - 5x^3 - 11x + 3 = 0 \]

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