Find the polynomial equation whose roots are the negatives of the roots of the equation
`x^4 -6x +7x^2 -2x +1=0`

Answer

To find the polynomial equation whose roots are the negatives of the roots of the equation \( x^4 - 6x + 7x^2 - 2x + 1 = 0 \), we can follow these steps:

### Step 1: Write the given polynomial
The given polynomial is:
\[ f(x) = x^4 + 7x^2 - 6x - 2x + 1 = 0 \]
This simplifies to:
\[ f(x) = x^4 + 7x^2 - 8x + 1 = 0 \]

### Step 2: Substitute \( x \) with \( -x \)
To find the polynomial whose roots are the negatives of the roots of \( f(x) \), we substitute \( x \) with \( -x \):
\[ f(-x) = (-x)^4 + 7(-x)^2 - 8(-x) + 1 \]

### Step 3: Simplify the expression
Now, simplify \( f(-x) \):
\[ f(-x) = x^4 + 7x^2 + 8x + 1 \]

### Step 4: Write the final polynomial equation
Thus, the polynomial equation whose roots are the negatives of the roots of the original polynomial is:
\[ x^4 + 7x^2 + 8x + 1 = 0 \]

### Summary
The required polynomial equation is:
\[ x^4 + 7x^2 + 8x + 1 = 0 \]