If −√2 and √3 are roots of the equation a₀ + a₁x + a₂x² + a₃x³ + x⁴ = 0 where a₀, a₁, a₂, a₃ are integers, then which one of the following is correct?
A.a₂ = a₃ = 0
B.a₂ = 0 and a₃ = −5
C.a₀ = 6, a₃ = 0✓ Correct
D.a₁ = 0 and a₂ = 5
Explanation
Since coefficients are integers and −√2, √3 are roots, their conjugates √2 and −√3 must also be roots. The polynomial is (x²−2)(x²−3) = x⁴ − 5x² + 6. So a₀ = 6, a₁ = 0, a₂ = −5, a₃ = 0. Therefore a₀ = 6 and a₃ = 0.
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