For a cubic equation with real coefficients, which patterns of roots are possible?

For a cubic equation with real coefficients, which patterns of roots are possible?

• A. Three real roots
• B. One real root and a complex conjugate pair
• C. Both of the above
• D. No real roots

Answer: (C)

For a cubic with real coefficients, non-real roots must come in complex conjugate pairs, and because the degree is odd, there must be at least one real root. That means you can have either all roots real (they may be distinct or some repeated) or exactly one real root with the other two forming a complex conjugate pair. A cubic cannot have no real roots. The discriminant of the cubic helps: it’s positive when there are three distinct real roots, negative when there is one real root and a complex pair, and zero when a root is repeated. So both patterns—three real roots or one real root plus a complex conjugate pair—are possible.

For a cubic with real coefficients, non-real roots must come in complex conjugate pairs, and because the degree is odd, there must be at least one real root. That means you can have either all roots real (they may be distinct or some repeated) or exactly one real root with the other two forming a complex conjugate pair. A cubic cannot have no real roots. The discriminant of the cubic helps: it’s positive when there are three distinct real roots, negative when there is one real root and a complex pair, and zero when a root is repeated. So both patterns—three real roots or one real root plus a complex conjugate pair—are possible.