If the zeros of a quadratic polynomial are (5+\sqrt{2}) and (5-\sqrt{2}), what is the polynomial in standard form?
Answer and explanation
Correct answer: x^2 - 10x + 23
Sum of zeros = (5+\sqrt{2})+(5-\sqrt{2}) = 10 and product = (5+\sqrt{2})(5-\sqrt{2}) = 5^2-(\sqrt{2})^2 = 25-2 = 23. For a monic quadratic with zeros α and β the polynomial is x^2 - (α+β)x + αβ. Hence the polynomial is x^2 - 10x + 23. The closest distractor C (x^2 - 10x + 27) only miscalculates the product; B and D have the wrong sign for the sum-term. Exam tip: conjugate irrational zeros give rational coefficients—use sum and product formulas directly to form the polynomial.
Frequently asked questions
What is the correct answer to this question?
x^2 - 10x + 23
Why is this the correct answer?
Sum of zeros = (5+\sqrt{2})+(5-\sqrt{2}) = 10 and product = (5+\sqrt{2})(5-\sqrt{2}) = 5^2-(\sqrt{2})^2 = 25-2 = 23. For a monic quadratic with zeros α and β the polynomial is x^2 - (α+β)x + αβ. Hence the polynomial is x^2 - 10x + 23. The closest distractor C (x^2 - 10x + 27) only miscalculates the product; B and D have the wrong sign for the sum-term. Exam tip: conjugate irrational zeros give rational coefficients—use sum and product formulas directly to form the polynomial.
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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