Which polynomial has zeros \(1+\sqrt{2}\) and \(1-\sqrt{2}\)?

\(x^2-2x-1\)

The sum of the roots is \((1+\sqrt{2})+(1-\sqrt{2})=2\) and the product is \((1+\sqrt{2})(1-\sqrt{2})=1-2=-1\). For a monic quadratic with roots \(\alpha\) and \(\beta\) the polynomial is \(x^2-(\alpha+\beta)x+\alpha\beta\). Thus the required polynomial is \(x^2-2x-1\). Closest distractor \(x^2+2x-1\) has sum \(-2\) (roots \(-1\pm\sqrt{2}\)), so it does not match; \(x^2-2x+1=(x-1)^2\) has a double root 1, and \(x^2+x-2\) has different sum/product. Exam tip: compute sum and product of given roots first, then plug into \(x^2-(\text{sum})x+(\text{product})\).

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