Find the equation whose roots are the reciprocals of the roots of
`x^4 + 3x^3 -6x^2 +2x -4=0`

Answer

To find the equation whose roots are the reciprocals of the roots of the given polynomial \( x^4 + 3x^3 - 6x^2 + 2x - 4 = 0 \), we can follow these steps:

### Step 1: Define the original polynomial
Let the original polynomial be defined as:
\[ f(x) = x^4 + 3x^3 - 6x^2 + 2x - 4 \]

### Step 2: Substitute \( x \) with \( \frac{1}{x} \)
To find the polynomial whose roots are the reciprocals of the roots of \( f(x) \), we substitute \( x \) with \( \frac{1}{x} \) in \( f(x) \):
\[ f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^4 + 3\left(\frac{1}{x}\right)^3 - 6\left(\frac{1}{x}\right)^2 + 2\left(\frac{1}{x}\right) - 4 \]

### Step 3: Simplify the expression
This simplifies to:
\[ f\left(\frac{1}{x}\right) = \frac{1}{x^4} + \frac{3}{x^3} - \frac{6}{x^2} + \frac{2}{x} - 4 \]

### Step 4: Multiply through by \( x^4 \)
To eliminate the fractions, multiply the entire equation by \( x^4 \):
\[ x^4 \cdot f\left(\frac{1}{x}\right) = 1 + 3x - 6x^2 + 2x^3 - 4x^4 \]

This gives us:
\[ 1 + 3x - 6x^2 + 2x^3 - 4x^4 = 0 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ -4x^4 + 2x^3 - 6x^2 + 3x + 1 = 0 \]

### Step 6: Multiply by -1
To write the polynomial in standard form, we multiply the entire equation by -1:
\[ 4x^4 - 2x^3 + 6x^2 - 3x - 1 = 0 \]

### Final Result
Thus, the equation whose roots are the reciprocals of the roots of the original polynomial is:
\[ 4x^4 - 2x^3 + 6x^2 - 3x - 1 = 0 \]

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