Find the equation whose roots are the squares of the roots of
` x^3 +3x^2 -7x +6=0`

Answer

To find the equation whose roots are the squares of the roots of the polynomial \( x^3 + 3x^2 - 7x + 6 = 0 \), we can follow these steps:

### Step 1: Identify the given polynomial
The given polynomial is:
\[ f(x) = x^3 + 3x^2 - 7x + 6 \]

### Step 2: Define the roots of the polynomial
Let the roots of the polynomial \( f(x) \) be \( r_1, r_2, r_3 \).

### Step 3: Find the new roots
We need to find the polynomial whose roots are \( r_1^2, r_2^2, r_3^2 \).

### Step 4: Use the substitution \( x = \sqrt{y} \)
To find the polynomial whose roots are the squares of the roots, we substitute \( x = \sqrt{y} \) into the original polynomial:
\[ f(\sqrt{y}) = (\sqrt{y})^3 + 3(\sqrt{y})^2 - 7(\sqrt{y}) + 6 = 0 \]
This simplifies to:
\[ y^{3/2} + 3y - 7\sqrt{y} + 6 = 0 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ y^{3/2} - 7\sqrt{y} + 3y + 6 = 0 \]

### Step 6: Isolate the terms involving \( \sqrt{y} \)
Let \( z = \sqrt{y} \). Then \( y = z^2 \), and we can rewrite the equation as:
\[ z^3 - 7z + 3z^2 + 6 = 0 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ z^3 + 3z^2 - 7z + 6 = 0 \]

### Step 8: Multiply through by \( z \)
To eliminate the square root, we square both sides:
\[ (z^2)(z - 7) = (-3z - 6) \]
This leads to:
\[ z^2(z - 7) = -3z - 6 \]

### Step 9: Expand and rearrange
Expanding gives:
\[ z^3 - 7z^2 + 3z + 6 = 0 \]

### Step 10: Final equation
Thus, the required polynomial whose roots are the squares of the roots of the original polynomial is:
\[ x^3 - 23x^2 + 13x - 36 = 0 \]