Multiple choice

Given that the complex number $z = -2 + 7i$ is a root to the equation: $z^3 + 6 z^2 + 61 z + 106 = 0$ find the real root to the equation.

  1. $ z = 2$
  2. $z = -2$
  3. $ z = -1$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since the coefficients are real, if -2 + 7i is a root, its conjugate -2 - 7i must also be a root. The quadratic factor is (z - (-2 + 7i))(z - (-2 - 7i)) = (z + 2)^2 + 49 = z^2 + 4z + 53. Dividing z^3 + 6z^2 + 61z + 106 by z^2 + 4z + 53 gives (z + 2). Thus, the real root is z = -2.

AI explanation

Since the coefficients of the polynomial are all real numbers, complex roots must occur in conjugate pairs, meaning the third root is the sum of the real root and the complex pair z and its conjugate. By Vieta's formulas, the sum of all three roots must equal -6. Since the sum of the given root and its conjugate is (-2 + 7i) + (-2 - 7i) = -4, the real root must be -6 - (-4), which is -2.

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