If the sum of the roots is -p and product of the roots `1/p` is, then the quadratic polynomial is:

Options

A. `k(-px^2+x/p+1)`
B. `k(-px^2+x/p-1)`
C. `(x^2+px+1/p)`
D. `k(x^2-px+x/p)`

Answer

To find the quadratic polynomial given the sum and product of its roots, we can use the standard form of a quadratic polynomial with roots \(\alpha\) and \(\beta\):

\[ P(x) = x^2 - (\alpha + \beta)x + \alpha\beta \]

Given:
- Sum of the roots, \(\alpha + \beta = -p\)
- Product of the roots, \(\alpha\beta = \frac{1}{p}\)

Let's substitute these values into the standard form of the quadratic polynomial.

### Step-by-Step Solution:

1. **Write the general form of the quadratic polynomial:**
\[ P(x) = x^2 - (\alpha + \beta)x + \alpha\beta \]

2. **Substitute the given sum of the roots:**
\[ \alpha + \beta = -p \]
So, the polynomial becomes:
\[ P(x) = x^2 - (-p)x + \alpha\beta \]
Simplifying the middle term:
\[ P(x) = x^2 + px + \alpha\beta \]

3. **Substitute the given product of the roots:**
\[ \alpha\beta = \frac{1}{p} \]
So, the polynomial becomes:
\[ P(x) = x^2 + px + \frac{1}{p} \]

Thus, the quadratic polynomial is:
\[ P(x) = x^2 + px + \frac{1}{p} \]