Find the product of the zeroes of the quadratic polynomial `x^2 - 3`.

Answer

To find the product of the zeroes of the quadratic polynomial \( x^2 - 3 \), we can follow these steps:

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

### Step 2: Set the polynomial equal to zero
To find the zeroes (roots) of the polynomial, we set it equal to zero:
\[ x^2 - 3 = 0 \]

### Step 3: Solve for \( x \)
Now, we solve for \( x \):
\[ x^2 = 3 \]
Taking the square root of both sides, we get:
\[ x = \sqrt{3} \quad \text{or} \quad x = -\sqrt{3} \]

### Step 4: Identify the zeroes
The zeroes of the polynomial are:
\[ \alpha = \sqrt{3} \quad \text{and} \quad \beta = -\sqrt{3} \]

### Step 5: Find the product of the zeroes
The product of the zeroes \( \alpha \) and \( \beta \) is given by:
\[ \alpha \cdot \beta = \sqrt{3} \cdot (-\sqrt{3}) \]
Calculating this gives:
\[ \alpha \cdot \beta = -(\sqrt{3})^2 = -3 \]

### Final Answer
Thus, the product of the zeroes of the quadratic polynomial \( x^2 - 3 \) is:
\[ \boxed{-3} \]

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