Find the equation whose roots are negative of the roots of the equation `x^(3)-3x^2+x+1=0`

Options

A. `x^3+3x^2+x-1=0`
B. `x^3-3x^2+x-1=0`
C. `x^3+3x^2-x-1=0`
D. `-x^3+3x^2+x+1=0`

Answer

To find the equation whose roots are the negatives of the roots of the equation \( x^3 - 3x^2 + x + 1 = 0 \), we can follow these steps:

### Step 1: Identify the given equation
The given equation is:
\[ x^3 - 3x^2 + x + 1 = 0 \]

### Step 2: Replace \( x \) with \( -x \)
To find the new equation whose roots are the negatives of the original roots, we replace \( x \) with \( -x \) in the original equation:
\[ (-x)^3 - 3(-x)^2 + (-x) + 1 = 0 \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ -x^3 - 3x^2 - x + 1 = 0 \]

### Step 4: Multiply through by -1
To make the leading coefficient positive, multiply the entire equation by -1:
\[ x^3 + 3x^2 + x - 1 = 0 \]

### Conclusion
The equation whose roots are the negatives of the roots of the original equation is:
\[ x^3 + 3x^2 + x - 1 = 0 \]