Question

Find a quadratic polynomial whose zeroes are \((5 - 2\sqrt{3})\) and \((5 + 2\sqrt{3})\).

Hint:

Irrational zeroes always occur in conjugate pairs for polynomials with rational coefficients. If one zero is \(a - \sqrt{b}\), the other must be \(a + \sqrt{b}\).
Sum is always \(2a\) and product is \(a^2 - b\).

Answer 1

Given:
The zeroes of the quadratic polynomial are: \[ \alpha = 5 - 2\sqrt{3}, \qquad \beta = 5 + 2\sqrt{3} \]

Step 1: Find sum of zeroes
\[ \alpha + \beta = (5 - 2\sqrt{3}) + (5 + 2\sqrt{3}) \] The irrational parts cancel out: \[ \alpha + \beta = 10 \]

Step 2: Find product of zeroes
\[ \alpha \beta = (5 - 2\sqrt{3})(5 + 2\sqrt{3}) \] Use identity \((a - b)(a + b) = a^2 - b^2\):
Here, \(a = 5\), \(b = 2\sqrt{3}\): \[ = 5^2 - (2\sqrt{3})^2 \] \[ = 25 - 4 \cdot 3 \] \[ = 25 - 12 = 13 \]

Step 3: Form the quadratic polynomial
A quadratic polynomial with given zeroes is: \[ x^2 - (\alpha + \beta)x + \alpha\beta \] Substitute values: \[ x^2 - 10x + 13 \]

Final Answer:
The required quadratic polynomial is:
\[ \boxed{x^2 - 10x + 13} \]