Question

The sum of the complex roots of the equation \[ x^4 - 2x^3 + x - 380 = 0 \] is:

• A. \(-3i + 3\)
• B. \(3i - 3\)
• C. \(-1\)
• D. \(1\)

Hint:

Use Vieta’s formula to separate sum of real roots and sum of complex roots in polynomials.

Answer 1

The Correct Option is D

Step 1: Identify roots.
The equation is a quartic. Suppose it has two real roots \(\alpha, \beta\) and two complex roots \(z_1, z_2\).

Step 2: Use Vieta's formula.
For a quartic \(x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0\), the sum of roots: \[ \alpha + \beta + z_1 + z_2 = -\frac{a_3}{1} = 2 \]

Step 3: Sum of complex roots.
Sum of complex roots: \[ z_1 + z_2 = (\text{sum of all roots}) - (\text{sum of real roots}) = 2 - (\alpha + \beta) \] Assuming the sum of real roots \(\alpha + \beta = 1\), then: \[ z_1 + z_2 = 2 - 1 = 1 \]

Step 4: Final conclusion.
\[ \boxed{1} \]