Write a quadratic polynomial whose sum of zeros is `( -1/4 )` and product of zeros is `( 1/4 ) `

Answer

To find a quadratic polynomial whose sum of zeros is \(-\frac{1}{4}\) and product of zeros is \(\frac{1}{4}\), we can follow these steps:

### Step 1: Understand the standard form of a quadratic polynomial
The standard form of a quadratic polynomial is given by:
\[ f(x) = ax^2 + bx + c \]
where \(a\), \(b\), and \(c\) are constants.

### Step 2: Use the relationship between the coefficients and the zeros
For a quadratic polynomial with roots \(\alpha\) and \(\beta\):
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

Given:
- Sum of zeros \(\alpha + \beta = -\frac{1}{4}\)
- Product of zeros \(\alpha \beta = \frac{1}{4}\)

### Step 3: Set \(a = 1\) for simplicity
Assuming \(a = 1\), we can express \(b\) and \(c\) in terms of the sum and product of the roots:
- From the sum of roots:
\[ -\frac{b}{1} = -\frac{1}{4} \implies b = \frac{1}{4} \]
- From the product of roots:
\[ \frac{c}{1} = \frac{1}{4} \implies c = \frac{1}{4} \]

### Step 4: Write the polynomial using the values of \(b\) and \(c\)
Substituting \(a\), \(b\), and \(c\) into the polynomial:
\[ f(x) = 1x^2 + \frac{1}{4}x + \frac{1}{4} \]
This simplifies to:
\[ f(x) = x^2 + \frac{1}{4}x + \frac{1}{4} \]

### Step 5: Eliminate the fraction by multiplying through by 4
To eliminate the fractions, multiply the entire polynomial by 4:
\[ 4f(x) = 4(x^2 + \frac{1}{4}x + \frac{1}{4}) = 4x^2 + x + 1 \]

### Final Answer
Thus, the required quadratic polynomial is:
\[ \boxed{4x^2 + x + 1} \]