Assertion(A): The sum and product of the zeroes of a quadratic polynomial are `(-1)/4` and `1/4` respectively.
Then the quadratic polynomial is `4x^(2)+x+1`
Reason (R):The quadratic polynomial whose sum and product of zeroes are given is `x^(2)`-(Sum of zeroes) x+ product of zeroes.

Options

A. Both A and R are true and R is the correct explanation for a
B. Both A and R are true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true

Answer

To solve the problem, we need to verify the assertion (A) and the reason (R) provided regarding the quadratic polynomial.

### Step 1: Understand the assertion and reason
The assertion states that the sum and product of the zeroes of a quadratic polynomial are given as:
- Sum of zeroes (α + β) = -1/4
- Product of zeroes (αβ) = 1/4

The reason states that the quadratic polynomial can be expressed as:
\[ p(x) = x^2 - (sum \ of \ zeroes) \cdot x + (product \ of \ zeroes) \]

### Step 2: Write the general form of the quadratic polynomial
Using the reason, we can write the polynomial as:
\[ p(x) = x^2 - (α + β)x + αβ \]

### Step 3: Substitute the values of sum and product
Substituting the values of the sum and product into the polynomial:
- Sum of zeroes = -1/4
- Product of zeroes = 1/4

So, we have:
\[ p(x) = x^2 - \left(-\frac{1}{4}\right)x + \frac{1}{4} \]
\[ p(x) = x^2 + \frac{1}{4}x + \frac{1}{4} \]

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

### Step 5: Conclusion
Thus, the quadratic polynomial is:
\[ 4x^2 + x + 1 \]

Now, we can conclude:
- The assertion (A) is true because we derived the polynomial as stated.
- The reason (R) is also true as it correctly describes how to form the polynomial from the sum and product of the zeroes.

### Final Answer
Both assertion (A) and reason (R) are true, and R is the correct explanation for A.

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