Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`sqrt(2),(1)/(3)`

Answer

To find a quadratic polynomial with given numbers as the sum and product of its zeroes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Sum of the zeroes (α + β) = \( \sqrt{2} \)
- Product of the zeroes (αβ) = \( \frac{1}{3} \)

2. **Use the Standard Form of a Quadratic Polynomial**:
The standard form of a quadratic polynomial based on its zeroes is given by:
\[ P(x) = x^2 - (α + β)x + (αβ) \]

3. **Substitute the Given Values into the Formula**:
Substitute the sum and product of the zeroes into the formula:
\[ P(x) = x^2 - (\sqrt{2})x + \left(\frac{1}{3}\right) \]

4. **Eliminate the Fraction**:
To eliminate the fraction, we can multiply the entire polynomial by 3 (the denominator of the product):
\[ P(x) = 3\left(x^2 - \sqrt{2}x + \frac{1}{3}\right) \]
This simplifies to:
\[ P(x) = 3x^2 - 3\sqrt{2}x + 1 \]

5. **Final Form of the Quadratic Polynomial**:
Thus, the required quadratic polynomial is:
\[ P(x) = 3x^2 - 3\sqrt{2}x + 1 \]

### Summary:
The quadratic polynomial with the given sum and product of its zeroes is:
\[ P(x) = 3x^2 - 3\sqrt{2}x + 1 \]