If `alpha, beta` are the zeroes of a polynomial such that `alpha+beta=6` and `alphabeta=4`, then write down the polynomial.

Options

A. `x^2-6x+4`
B. `x^2+6x+4`
C. `x^2+4x+6`
D. `x^2-4x+6`

Answer

To find the polynomial with given zeros \( \alpha \) and \( \beta \) such that \( \alpha + \beta = 6 \) and \( \alpha \beta = 4 \), we can follow these steps:

### Step 1: Use the relationship between zeros and coefficients
The polynomial can be expressed in the standard form as:
\[ P(x) = x^2 - (\alpha + \beta)x + \alpha \beta \]

### Step 2: Substitute the values of the sum and product of the roots
From the problem, we know:
- \( \alpha + \beta = 6 \)
- \( \alpha \beta = 4 \)

Substituting these values into the polynomial gives:
\[ P(x) = x^2 - (6)x + 4 \]

### Step 3: Write the polynomial in standard form
Now, simplifying this expression, we get:
\[ P(x) = x^2 - 6x + 4 \]

### Final Answer:
Thus, the polynomial is:
\[ P(x) = x^2 - 6x + 4 \]

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