A quadratic polynomial with sun and product of zeroes as - `(1)/(4) and (1)/(4)`, respectively, is:

Options

A. `4x^(2) - x + 1`
B. `4x^(2) + x + 1`
C. `4x^(2) + x - 1`
D. `4x^(2) - x - 1`

Answer

To find the quadratic polynomial given the sum and product of its zeroes, we can use the standard form of a quadratic polynomial, which is:

\[ P(x) = x^2 - (sum \ of \ zeroes) \cdot x + (product \ of \ zeroes) \]

Given:
- Sum of zeroes = \( -\frac{1}{4} \)
- Product of zeroes = \( \frac{1}{4} \)

### Step 1: Write the polynomial using the sum and product of zeroes.

Substituting the values into the polynomial form:

\[ P(x) = x^2 - \left(-\frac{1}{4}\right) \cdot x + \frac{1}{4} \]

### Step 2: Simplify the polynomial.

This simplifies to:

\[ P(x) = x^2 + \frac{1}{4}x + \frac{1}{4} \]

### Step 3: Eliminate the fraction by multiplying through by 4.

To eliminate the fraction, multiply the entire equation by 4:

\[ 4P(x) = 4x^2 + 4 \cdot \frac{1}{4}x + 4 \cdot \frac{1}{4} \]

This simplifies to:

\[ 4P(x) = 4x^2 + x + 1 \]

### Step 4: Write the final polynomial.

Thus, the quadratic polynomial is:

\[ P(x) = 4x^2 + x + 1 \]

### Final Answer:

The required quadratic polynomial is:

\[ 4x^2 + x + 1 \]

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