Calculate the vlaue of `alpha^(2) - beta^(2)` where `alpha, beta` are zeroes of the polynomial `x^(2) - 5x + 6`.

Options

A. 0
B. 2
C. 7
D. 5

Answer

To solve the problem, we need to calculate the value of \( \alpha^2 - \beta^2 \) where \( \alpha \) and \( \beta \) are the roots (zeroes) of the polynomial \( x^2 - 5x + 6 \).

### Step 1: Find the roots of the polynomial
The roots of the polynomial \( x^2 - 5x + 6 \) can be found using the factorization method. We need to factor the quadratic expression.

The expression can be factored as:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 2: Identify the roots
From the factorization, we can identify the roots:
\[ \alpha = 2 \quad \text{and} \quad \beta = 3 \]

### Step 3: Calculate \( \alpha^2 - \beta^2 \)
We can use the difference of squares formula, which states that:
\[ \alpha^2 - \beta^2 = (\alpha - \beta)(\alpha + \beta) \]

First, we calculate \( \alpha - \beta \) and \( \alpha + \beta \):
\[ \alpha - \beta = 2 - 3 = -1 \]
\[ \alpha + \beta = 2 + 3 = 5 \]

Now, we can substitute these values into the difference of squares formula:
\[ \alpha^2 - \beta^2 = (-1)(5) = -5 \]

### Final Answer
Thus, the value of \( \alpha^2 - \beta^2 \) is:
\[ \boxed{-5} \]