Multiple choice

The equation of the lowest degree with rational coefficients having a root $1+i$ is

  1. $x^{4}+4=0$
  2. $x^{4}-4=0$
  3. $x^{2}-2x+2=0$
  4. $x^{2}+4x+1=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If 1+i is a root, then 1-i must also be a root for rational coefficients. The quadratic is (x - (1+i))(x - (1-i)) = ((x-1) - i)((x-1) + i) = (x-1)^2 - i^2 = x^2 - 2x + 1 + 1 = x^2 - 2x + 2 = 0.

AI explanation

Since the polynomial must have rational coefficients, the complex conjugate 1 - i must also be a root. The sum of the roots is (1 + i) + (1 - i) = 2, and the product of the roots is (1 + i)(1 - i) = 1 + 1 = 2. Forming the quadratic equation using Vieta's formulas gives x^2 - (sum)x + (product) = 0. Therefore, the required equation is x^2 - 2x + 2 = 0.

Keep practicing — related questions