Multiple choice

The quadratic polynomial whose zeros are $3+\sqrt{5}$ and $3-\sqrt{5}$ is

  1. $y^2-6y+4$
  2. $2y^2-6y+4$
  3. $2y^2-12y+8$
  4. $y^2+10y+4$
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A Correct answer
Explanation

The sum of the zeros is (3+sqrt(5)) + (3-sqrt(5)) = 6, and the product is (3+sqrt(5))(3-sqrt(5)) = 9 - 5 = 4. A quadratic polynomial is given by y^2 - (sum)y + (product), which results in y^2 - 6y + 4.

AI explanation

If the zeros of a quadratic polynomial are alpha and beta, the polynomial can be written as y^2 - (alpha + beta)y + (alpha * beta). The sum of the zeros is (3 + sqrt(5)) + (3 - sqrt(5)) = 6, and the product is (3)^2 - (sqrt(5))^2 = 9 - 5 = 4. Substituting these values gives the polynomial y^2 - 6y + 4. The required polynomial is y^2 - 6y + 4.

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