Which quadratic polynomial has zeros \(2+\sqrt{10}\) and \(2-\sqrt{10}\)?
Answer and explanation
Correct answer: \(x^2-4x-6\)
The sum of zeros is \((2+\sqrt{10})+(2-\sqrt{10})=4\) and the product is \((2+\sqrt{10})(2-\sqrt{10})=4-10=-6\). For a monic quadratic the polynomial is \(x^2-(\text{sum})x+\text{product}\), so \(x^2-4x-6\) is correct. Closest distractors fail because: \(x^2-4x+6\) has the wrong constant term (+6 instead of -6), \(x^2+4x-6\) has the wrong sign for the linear term, and \(x^2-2x-10\) has neither sum nor product matching. Exam tip: compute sum and product first and form \(x^2-({\text{sum}})x+{\text{product}}\).
Frequently asked questions
What is the correct answer to this question?
\(x^2-4x-6\)
Why is this the correct answer?
The sum of zeros is \((2+\sqrt{10})+(2-\sqrt{10})=4\) and the product is \((2+\sqrt{10})(2-\sqrt{10})=4-10=-6\). For a monic quadratic the polynomial is \(x^2-(\text{sum})x+\text{product}\), so \(x^2-4x-6\) is correct. Closest distractors fail because: \(x^2-4x+6\) has the wrong constant term (+6 instead of -6), \(x^2+4x-6\) has the wrong sign for the linear term, and \(x^2-2x-10\) has neither sum nor product matching. Exam tip: compute sum and product first and form \(x^2-({\text{sum}})x+{\text{product}}\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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