In a cubic polynomial, what condition must be met for it to have real coefficients when two roots are complex?

In a cubic polynomial, when the polynomial has real coefficients, any complex roots must indeed occur in conjugate pairs. This means if one root is a complex number, say ( a + bi ), then its conjugate ( a - bi ) must also be a root of the polynomial. This is a fundamental property of polynomials with real coefficients due to the way polynomial equations are structured.

When complex roots appear, they cannot exist alone; if they did, the coefficients of the polynomial would not remain entirely real after factoring. Therefore, the presence of one complex root necessitates the presence of its conjugate to ensure that the resulting polynomial does not end up with imaginary coefficients.

In this scenario, since a cubic polynomial has three roots, if it has two complex roots (which must be a conjugate pair), the third root must be real for the total number of roots to match the degree of the polynomial. This justification aligns perfectly with the condition needed for maintaining real coefficients within the polynomial.

The other options do not accurately describe the necessary condition for real coefficients in this context. For instance, having all real roots would indeed result in real coefficients, but it does not relate to the specifics regarding complex roots. Similarly, there is no requirement related