Multiple choice

The quadratic polynomial whose zeroes are $(5+ \sqrt {2}) and ( 5- \sqrt{2})$ is

  1. $x^{2} -10x+23$
  2. $x^{2}+x+1$
  3. $x^{2}+x+2$
  4. $x^{2}+x+3$
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A Correct answer
Explanation

Sum of roots = (5+sqrt(2)) + (5-sqrt(2)) = 10. Product of roots = (5+sqrt(2))(5-sqrt(2)) = 25 - 2 = 23. The quadratic is x^2 - (sum)x + (product) = x^2 - 10x + 23.

AI explanation

Using the relationship between roots and coefficients, the required quadratic polynomial is x^2 - (sum of zeroes)x + product of zeroes. The sum of the roots is 5 + sqrt(2) + 5 - sqrt(2) = 10, and the product is (5 + sqrt(2))(5 - sqrt(2)) = 25 - 2 = 23. Substituting these values gives the polynomial x^2 - 10x + 23.

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