Multiple choice

The equation of the lowest degree with rational coefficients having a root $\sqrt{7}-i$, is

  1. $x^{4}-16x^{2}+36=0$
  2. $x^{4}-12x^{2}+64=0$
  3. $x^{4}-16x^{2}+64=0$
  4. $x^{4}-12x^{2}+36=0$
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B Correct answer
AI explanation

For a polynomial to have rational coefficients, any complex root sqrt(7)-i must be paired with its conjugate sqrt(7)+i, and any irrational root must be paired with its conjugate -sqrt(7)-i and -sqrt(7)+i. The required polynomial is the product of the factors (x - (sqrt(7)-i))(x - (sqrt(7)+i))(x - (-sqrt(7)-i))(x - (-sqrt(7)+i)). Grouping the conjugate pairs gives ((x-sqrt(7))^2 + 1)((x+sqrt(7))^2 + 1). Expanding the first group yields x^2 - 2sqrt(7)x + 8, and expanding the second yields x^2 + 2sqrt(7)x + 8. Multiplying these quadratics gives (x^2+8)^2 - (2sqrt(7)x)^2 = 0. This expands to x^4 + 16x^2 + 64 - 28x^2 = 0, which simplifies to the equation x^4 - 12x^2 + 64 = 0.

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