Find a quadratic polynomial, the sum of whose zeroes is `(5)/(3)` and product is 2.

Explanation

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

### Step 1: Identify the sum and product of the zeroes
We are given:
- Sum of the zeroes (α + β) = \( \frac{5}{3} \)
- Product of the zeroes (αβ) = 2

### Step 2: Use the standard form of a quadratic polynomial
The standard form of a quadratic polynomial with zeroes α and β can be expressed as:
\[ P(x) = x^2 - (α + β)x + (αβ) \]

### Step 3: Substitute the values of sum and product into the polynomial
Substituting the values we have:
\[ P(x) = x^2 - \left(\frac{5}{3}\right)x + 2 \]

### Step 4: Eliminate the fraction by multiplying through by 3
To eliminate the fraction, we can multiply the entire equation by 3:
\[ 3P(x) = 3\left(x^2 - \left(\frac{5}{3}\right)x + 2\right) \]
This simplifies to:
\[ 3P(x) = 3x^2 - 5x + 6 \]

### Step 5: Write the final polynomial
Thus, the quadratic polynomial we are looking for is:
\[ P(x) = 3x^2 - 5x + 6 \]

### Summary
The quadratic polynomial with the given sum and product of zeroes is:
\[ P(x) = 3x^2 - 5x + 6 \]

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