If `alpha` and `beta` are the zeros of the quadratic polynomial `f(x) = 3x^2 - 7x - 6`, find a polynomial whose zeros are `alpha^(2)` and `beta^(2)`

Answer

To find a polynomial whose zeros are \( \alpha^2 \) and \( \beta^2 \), where \( \alpha \) and \( \beta \) are the zeros of the polynomial \( f(x) = 3x^2 - 7x - 6 \), we can follow these steps:

### Step 1: Identify the coefficients and find the sum and product of the roots
The given polynomial is:
\[ f(x) = 3x^2 - 7x - 6 \]
From the polynomial, we can identify:
- \( a = 3 \)
- \( b = -7 \)
- \( c = -6 \)

Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{-7}{3} = \frac{7}{3} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{-6}{3} = -2 \)

### Step 2: Calculate the sum of the squares of the roots
To find the polynomial whose zeros are \( \alpha^2 \) and \( \beta^2 \), we need to calculate:
- The sum of the squares of the roots \( \alpha^2 + \beta^2 \).

Using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

Substituting the values we found:
\[ \alpha^2 + \beta^2 = \left(\frac{7}{3}\right)^2 - 2(-2) \]
Calculating this:
\[ \alpha^2 + \beta^2 = \frac{49}{9} + 4 = \frac{49}{9} + \frac{36}{9} = \frac{85}{9} \]

### Step 3: Calculate the product of the squares of the roots
Next, we calculate the product of the squares of the roots:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 \]
Substituting the value of \( \alpha \beta \):
\[ \alpha^2 \beta^2 = (-2)^2 = 4 \]

### Step 4: Form the new polynomial
Now that we have the sum and product of the new roots \( \alpha^2 \) and \( \beta^2 \):
- Sum of the roots \( \alpha^2 + \beta^2 = \frac{85}{9} \)
- Product of the roots \( \alpha^2 \beta^2 = 4 \)

The polynomial with roots \( \alpha^2 \) and \( \beta^2 \) can be expressed as:
\[ x^2 - (\alpha^2 + \beta^2)x + \alpha^2 \beta^2 = 0 \]
Substituting the values we found:
\[ x^2 - \frac{85}{9}x + 4 = 0 \]

### Step 5: Clear the fraction
To eliminate the fraction, we can multiply the entire equation by 9:
\[ 9x^2 - 85x + 36 = 0 \]

### Final Answer
Thus, the polynomial whose zeros are \( \alpha^2 \) and \( \beta^2 \) is:
\[ 9x^2 - 85x + 36 = 0 \]