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

Answer

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

### Step 1: Identify the given values
We are given:
- Sum of the zeroes (α + β) = \( \frac{1}{4} \)
- Product of the zeroes (αβ) = -1

### Step 2: Write the general form of a quadratic polynomial
The general form of a quadratic polynomial can be expressed as:
\[ P(x) = x^2 - (α + β)x + (αβ) \]
Substituting the values we have:
\[ P(x) = x^2 - \left(\frac{1}{4}\right)x + (-1) \]

### Step 3: Substitute the values into the polynomial
Now we substitute the values into the polynomial:
\[ P(x) = x^2 - \frac{1}{4}x - 1 \]

### Step 4: Eliminate the fraction by multiplying through by 4
To eliminate the fraction, we can multiply the entire equation by 4:
\[ 4P(x) = 4\left(x^2 - \frac{1}{4}x - 1\right) \]
This gives us:
\[ 4P(x) = 4x^2 - x - 4 \]

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

### Final Answer:
The quadratic polynomial is:
\[ 4x^2 - x - 4 \]

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