Find the equation whose roots are the reciprocals of the roots of
` x^4 -3x^3 +7x^2 +5x -2=0`

Answer

To find the equation whose roots are the reciprocals of the roots of the polynomial \( x^4 - 3x^3 + 7x^2 + 5x - 2 = 0 \), we can follow these steps:

### Step 1: Define the original polynomial
Let \( f(x) = x^4 - 3x^3 + 7x^2 + 5x - 2 \).

### Step 2: Substitute \( x \) with \( \frac{1}{x} \)
To find the equation whose roots are the reciprocals of the roots of \( f(x) \), we substitute \( x \) with \( \frac{1}{x} \) in the polynomial:
\[ f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^4 - 3\left(\frac{1}{x}\right)^3 + 7\left(\frac{1}{x}\right)^2 + 5\left(\frac{1}{x}\right) - 2 = 0 \]

### Step 3: Simplify the expression
This gives us:
\[ \frac{1}{x^4} - \frac{3}{x^3} + \frac{7}{x^2} + \frac{5}{x} - 2 = 0 \]

### Step 4: Multiply through by \( x^4 \)
To eliminate the fractions, multiply the entire equation by \( x^4 \):
\[ 1 - 3x + 7x^2 + 5x^3 - 2x^4 = 0 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ -2x^4 + 5x^3 + 7x^2 - 3x + 1 = 0 \]

### Step 6: Multiply by -1
To write the polynomial in standard form, we multiply the entire equation by -1:
\[ 2x^4 - 5x^3 - 7x^2 + 3x - 1 = 0 \]

### Final Result
Thus, the equation whose roots are the reciprocals of the roots of the original polynomial is:
\[ 2x^4 - 5x^3 - 7x^2 + 3x - 1 = 0 \]

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