Find a polynomial a function P of the lowest possible degree, having real coefficients, a leading coefficient of 1, and with the given zeros, 1+2i, -1, and 2

Question

How do I find the lowest degree with the 1+2i, -1, and 2

Answer 1

By congugate zeros theorem 1 - 2i is also zero.

Then P(x) = (x - 2)(x + 1)(x - 1 - 2i)(x - 1 + 2i) = (x² - x - 2)(x² - 2x + 1 + 4) = (x² - x - 2)(x² - 2x + 5) =

x⁴ - 2x³ + 5x² - x³ + 2x² + 5x - 2x² + 4x - 10 = x⁴ - 3x³ + 5x² + 9x - 10