If -\sqrt{2} and \sqrt{3} are roots of the equation a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+x^{4}=0 where a_{0}, a_{1}, a_{2}, a_{3} are integers, then which one of the following is correct?

  1. A. a_{2}=a_{3}=0
  2. B. a_{2}=0 and a_{3}=-5
  3. C. a_{0}=6, a_{3}=0 ✓
  4. D. a_{1}=0 and a_{2}=5

Correct Answer: C. a_{0}=6, a_{3}=0

Explanation

Since the coefficients are integers, the irrational roots must occur in conjugate pairs. So the other roots are \sqrt{2} and -\sqrt{3}. The polynomial is (x-\sqrt{2})(x+\sqrt{2})(x-\sqrt{3})(x+\sqrt{3}) = (x^2-2)(x^2-3) = x^4 - 5x^2 + 6. Comparing this to the given equation, a_0 = 6, a_1 = 0, a_2 = -5, a_3 = 0. Hence, a_0=6 and a_3=0.

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