Multiple choice

Consider the following statements: I. The equation of lowest degree with rational coefficients, one of whose roots is $\sqrt{7}+\sqrt{3}$ is $x^{4}-20x^{2}+16=0$. II. The equation of lowest degree with rational coefficients, one of whose roots is $\sqrt{2}+i\sqrt{3}$ is $x^{4}+2x^{2}+25=0$.

  1. only I is true

  2. only II is true

  3. both I and II are true

  4. neither I nor II ture

Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For I, if x = sqrt(7) + sqrt(3), then x^2 = 7 + 3 + 2*sqrt(21) = 10 + 2*sqrt(21). (x^2 - 10)^2 = 4*21 = 84, so x^4 - 20x^2 + 100 = 84, giving x^4 - 20x^2 + 16 = 0. For II, if x = sqrt(2) + i*sqrt(3), x^2 = 2 - 3 + 2*i*sqrt(6) = -1 + 2*i*sqrt(6). (x^2 + 1)^2 = -24, so x^4 + 2x^2 + 1 + 24 = 0, giving x^4 + 2x^2 + 25 = 0.