Mathematics

This problem hinges on the conjugate root theorem: when a polynomial has real coefficients, complex roots always come in conjugate pairs. Since $x = 1 + i$ is a root, you know $x = 1 - i$ must also be a root. Rather than expanding $(x - (1+i))(x - (1-i))$ and dividing, notice that multiplying these two factors gives you a real quadratic—use that to factor out a quadratic from the cubic and find the remaining root.

This problem hinges on the conjugate root theorem: when a polynomial has real coefficients, complex roots always come in conjugate pairs. Since $x = 1 + i$ is a root, you know $x = 1 - i$ must also be a root. Rather than expanding $(x - (1+i))(x - (1-i))$ and dividing, notice that multiplying these two factors gives you a real quadratic—use that to factor out a quadratic from the cubic and find the remaining root.