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

Answer

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

### Step 1: Understand the relationship between roots and coefficients
For a quadratic polynomial of the form \( ax^2 + bx + c \), if the roots are \( r_1 \) and \( r_2 \):
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)

### Step 2: Identify the given values
From the problem:
- The sum of the roots is given as \( -p \).
- The product of the roots is given as \( \frac{1}{p} \).

### Step 3: Set up the polynomial using the relationships
Using the relationships from Step 1, we can express the coefficients of the polynomial:
- Since the sum of the roots \( r_1 + r_2 = -p \), we have:
\[ -\frac{b}{a} = -p \implies b = ap \]
- Since the product of the roots \( r_1 \cdot r_2 = \frac{1}{p} \), we have:
\[ \frac{c}{a} = \frac{1}{p} \implies c = \frac{a}{p} \]

### Step 4: Assume \( a = 1 \) for simplicity
To find the polynomial, we can assume \( a = 1 \) (which is common for standard form):
- Then \( b = p \)
- And \( c = \frac{1}{p} \)

### Step 5: Write the quadratic polynomial
Substituting \( a \), \( b \), and \( c \) into the polynomial form:
\[ f(x) = ax^2 + bx + c = 1x^2 + px + \frac{1}{p} \]
Thus, the quadratic polynomial is:
\[ f(x) = x^2 + px + \frac{1}{p} \]

### Conclusion
The quadratic polynomial that meets the given conditions is:
\[ x^2 + px + \frac{1}{p} \]

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