If a and B are the zeroes of the polynomial f(x) = `x^(2) - 4x - 5` then find the value of `alpha^(2) + beta^(2)`.

Answer

To find the value of \( \alpha^2 + \beta^2 \) where \( \alpha \) and \( \beta \) are the zeros of the polynomial \( f(x) = x^2 - 4x - 5 \), we can follow these steps:

### Step 1: Identify the coefficients
The polynomial is given as:
\[ f(x) = x^2 - 4x - 5 \]
Here, we can identify the coefficients:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -4 \) (coefficient of \( x \))
- \( c = -5 \) (constant term)

### Step 2: Calculate the sum and product of the zeros
Using Vieta's formulas:
- The sum of the zeros \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-4}{1} = 4 \]
- The product of the zeros \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{-5}{1} = -5 \]

### Step 3: Use the identity to find \( \alpha^2 + \beta^2 \)
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (4)^2 - 2(-5) \]

### Step 4: Calculate \( \alpha^2 + \beta^2 \)
Now, calculate:
\[ \alpha^2 + \beta^2 = 16 + 10 = 26 \]

### Final Answer
Thus, the value of \( \alpha^2 + \beta^2 \) is:
\[ \boxed{26} \]