Given two roots of a cubic equation are complex, what is the standard form of the cubic equation with real coefficients?

To determine the standard form of a cubic equation with real coefficients when two of its roots are complex, we must recall that complex roots for polynomial equations with real coefficients always come in conjugate pairs. This means if one root is (a + bi) (where (a) and (b) are real numbers), the other root must be (a - bi).

Given that the cubic equation has three roots, if two are complex and conjugates of each other, the third root must necessarily be a real number.

The correct answer can be verified by confirming that the coefficients of the polynomial, when expanded from its roots, yield real values in the standard form. The structure is such that it would be ( (x - r)(x - (a + bi))(x - (a - bi)) ), where (r) is the real root.

The provided option, which is a polynomial ( x^3 - 6x^2 + x + 34 = 0 ), can be analyzed further by potentially factoring or verifying against roots. If it satisfies the necessary conditions when evaluated or if there exists a root that is real and leads to the resulting polynomial being a product of real linear factors multiplied by a quadratic