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: Hunt for a rational root of the quartic.
Testing \(x=5\) in \(x^4-2x^3+x-380=0\): \(625-250+5-380=0\), so \(x=5\) is a root.

Step 2: Divide it out and find the next root.
Dividing by \((x-5)\) gives the cubic \(x^3+3x^2+15x+76=0\).
Testing \(x=-4\) here: \(-64+48-60+76=0\), so \(x=-4\) is also a root.

Step 3: Divide again to isolate the complex pair.
Dividing \(x^3+3x^2+15x+76\) by \((x+4)\) leaves the quadratic \(x^2-x+19=0\).
Its discriminant is \(1-76=-75<0\), so its two roots are complex conjugates, and these are exactly the complex roots the question is asking about.

Step 4: Sum the complex roots.
For \(x^2-x+19=0\), the sum of roots is \(-\frac{-1}{1}=1\).
\[ \boxed{1} \]