Which of the following polynomials has zeros \,(\sqrt{2}+\sqrt{3}) and \,(\sqrt{2}-\sqrt{3})?

\(x^2-2\sqrt{2}\,x-1\)

Let the roots be \(\alpha=\sqrt{2}+\sqrt{3}\) and \(\beta=\sqrt{2}-\sqrt{3}\). Then \(\alpha+\beta=2\sqrt{2}\) and \(\alpha\beta=(\sqrt{2})^2-(\sqrt{3})^2=2-3=-1\). For a quadratic with these roots use \(x^2-(\text{sum})x+\text{product}\), giving \(x^2-2\sqrt{2}\,x-1\) (option A). Option B has the wrong sign on the linear term, so its sum of roots would be \(-2\sqrt{2}\); option C uses \(2\sqrt{3}\) which does not match the actual sum; option D equals \(0\) at ±\(\sqrt{5}\), not the given irrational roots. Exam tip: compute sum and product of the given roots first — this directly yields the required quadratic polynomial.

This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.