A group consists of 15 males and 12 females. Write a quadratic polynomial whose roots are equal to the number of males and females .

Options

A. `x^(2) - 27 x + 180`
B. `x^(2) + 27 x - 180`
C. `x^(2) + 3x - 27`
D. None of these

Answer

To find a quadratic polynomial whose roots are equal to the number of males (15) and females (12), we can follow these steps:

### Step 1: Identify the roots
The roots of the polynomial are given as:
- \( \alpha = 15 \) (number of males)
- \( \beta = 12 \) (number of females)

### Step 2: Use the standard form of a quadratic polynomial
The standard form of a quadratic polynomial with roots \( \alpha \) and \( \beta \) is given by:
\[ P(x) = (x - \alpha)(x - \beta) \]

### Step 3: Substitute the roots into the polynomial
Substituting \( \alpha \) and \( \beta \) into the polynomial gives:
\[ P(x) = (x - 15)(x - 12) \]

### Step 4: Expand the polynomial
Now, we will expand the expression:
\[ P(x) = x^2 - 12x - 15x + 180 \]
\[ P(x) = x^2 - 27x + 180 \]

### Final Polynomial
Thus, the quadratic polynomial whose roots are 15 and 12 is:
\[ P(x) = x^2 - 27x + 180 \]