Understand and use the complex conjugate root theorem for polynomials with real coefficients, e.g. factorise a cubic polynomial with real coefficients given one factor.

A cubic polynomial P(z) = z³ - 2z² + 4z - 8 has real coefficients. Given that z = 2i is a root of this polynomial, find all roots of P(z) and express P(z) as a product of linear factors.

The polynomial \(Q(z) = z^3 - 5z^2 + 11z - 15\) has real coefficients. Given that \(z = 2 + i\) is a root of \(Q(z)\), which of the following is another root of the polynomial?

A cubic polynomial with real coefficients has roots $3i$ and $5$. Which of the following is a factor of this polynomial?

A cubic polynomial P(x) with real coefficients has one root z = 3 + 2i. Which of the following must also be a root of P(x)?