Find the quadratic polynomial whose zeros are
`(5+2sqrt(3))` and `(5-2sqrt(3))`

Answer

To find the quadratic polynomial whose zeros are \( 5 + 2\sqrt{3} \) and \( 5 - 2\sqrt{3} \), we can follow these steps:

### Step 1: Identify the zeros
The zeros given are:
- \( \alpha = 5 + 2\sqrt{3} \)
- \( \beta = 5 - 2\sqrt{3} \)

### Step 2: Calculate the sum of the zeros
The sum of the zeros \( \alpha + \beta \) can be calculated as follows:
\[ \alpha + \beta = (5 + 2\sqrt{3}) + (5 - 2\sqrt{3}) = 5 + 5 + 2\sqrt{3} - 2\sqrt{3} = 10 \]

### Step 3: Calculate the product of the zeros
The product of the zeros \( \alpha \cdot \beta \) can be calculated as:
\[ \alpha \cdot \beta = (5 + 2\sqrt{3})(5 - 2\sqrt{3}) \]
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ = 5^2 - (2\sqrt{3})^2 = 25 - (4 \cdot 3) = 25 - 12 = 13 \]

### Step 4: Form the quadratic polynomial
The general form of a quadratic polynomial with zeros \( \alpha \) and \( \beta \) is given by:
\[ P(x) = x^2 - (\text{sum of zeros})x + (\text{product of zeros}) \]
Substituting the values we found:
\[ P(x) = x^2 - (10)x + 13 \]
Thus, the quadratic polynomial is:
\[ P(x) = x^2 - 10x + 13 \]

### Final Answer
The quadratic polynomial whose zeros are \( 5 + 2\sqrt{3} \) and \( 5 - 2\sqrt{3} \) is:
\[ P(x) = x^2 - 10x + 13 \]