If `alpha, beta` are the two zeros of the polynomial `f (y) = y^2 - 8y + a` and `alpha^(2) + beta^(2) = 40`, find the value of a.

Answer

To find the value of \( a \) in the polynomial \( f(y) = y^2 - 8y + a \) given that \( \alpha^2 + \beta^2 = 40 \), we can follow these steps:

### Step 1: Identify the sum and product of the roots
For a quadratic polynomial of the form \( y^2 + by + c \), the sum of the roots \( \alpha + \beta \) and the product of the roots \( \alpha \beta \) can be expressed as:
- Sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots: \( \alpha \beta = \frac{c}{a} \)

In our polynomial \( f(y) = y^2 - 8y + a \):
- Here, \( b = -8 \) and \( c = a \).
- Therefore, the sum of the roots is:
\[ \alpha + \beta = -\frac{-8}{1} = 8 \]
- The product of the roots is:
\[ \alpha \beta = \frac{a}{1} = a \]

### Step 2: Use the identity for the sum of squares of roots
We know that:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we have:
\[ \alpha^2 + \beta^2 = (8)^2 - 2(\alpha \beta) \]
\[ \alpha^2 + \beta^2 = 64 - 2a \]

### Step 3: Set up the equation with the given condition
We are given that \( \alpha^2 + \beta^2 = 40 \). Thus, we can set up the equation:
\[ 64 - 2a = 40 \]

### Step 4: Solve for \( a \)
Now, we can solve for \( a \):
\[ 64 - 40 = 2a \]
\[ 24 = 2a \]
\[ a = \frac{24}{2} = 12 \]

### Conclusion
The value of \( a \) is \( 12 \).