For a quartic equation with real coefficients, which is NOT a possible root pattern?

For a quartic equation with real coefficients, which is NOT a possible root pattern?

• A. Four real roots
• B. Two real roots and a complex conjugate pair
• C. Two sets of conjugate pairs
• D. One real root and a complex triple

Answer: (D)

Real coefficients force complex roots to come in conjugate pairs. In a quartic, that means the number of non-real roots must be even: 0, 2, or 4. So patterns with all real roots, or two real roots plus a complex conjugate pair, or two pairs of complex conjugates are possible. A real root with a complex triple would require three non-real roots, which cannot happen because non-real roots occur in pairs. Therefore that pattern cannot occur.

Real coefficients force complex roots to come in conjugate pairs. In a quartic, that means the number of non-real roots must be even: 0, 2, or 4. So patterns with all real roots, or two real roots plus a complex conjugate pair, or two pairs of complex conjugates are possible. A real root with a complex triple would require three non-real roots, which cannot happen because non-real roots occur in pairs. Therefore that pattern cannot occur.