Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`-(1)/(4),(1)/(4)`

Answer

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

### Step 1: Identify the sum and product of the zeroes
The sum of the zeroes (α + β) is given as \(-\frac{1}{4}\) and the product of the zeroes (αβ) is given as \(\frac{1}{4}\).

### Step 2: Use the standard form of a quadratic polynomial
The standard form of a quadratic polynomial based on the sum and product of its zeroes is given by:
\[ P(x) = x^2 - (α + β)x + (αβ) \]
Substituting the values of the sum and product into this formula, we get:
\[ P(x) = x^2 - \left(-\frac{1}{4}\right)x + \frac{1}{4} \]

### Step 3: Simplify the polynomial
This simplifies to:
\[ P(x) = x^2 + \frac{1}{4}x + \frac{1}{4} \]

### Step 4: Eliminate fractions by multiplying by a common denominator
To eliminate the fractions, we can multiply the entire polynomial by 4 (the least common multiple of the denominators):
\[ 4P(x) = 4\left(x^2 + \frac{1}{4}x + \frac{1}{4}\right) \]
Distributing the 4 gives:
\[ 4P(x) = 4x^2 + x + 1 \]

### Step 5: Write the final polynomial
Thus, the quadratic polynomial we are looking for is:
\[ P(x) = 4x^2 + x + 1 \]

### Summary
The quadratic polynomial with the given sum and product of its zeroes is:
\[ \boxed{4x^2 + x + 1} \]