A degree 4 polynomial P(x) with integer coefficient has zeros 2i and 4, with 4 being a zero of multiciplity 2. Moreover, the coefficient of x^4 is 1. Find the polynomial.

Answer 1

If 2i is a zero, then by the Conjugate Root Theorem, -2i is also a zero. The polynomial then (in factored form) is f(x) = A(x - 4)(x - 4)(x - 2i)(x + 2i) where A is the leading coefficient of the polynomial. They tell you that A = 1. So f(x) = (x - 4)(x - 4)(x - 2i)(x + 2i). Multiply this out and you get: f(x) = x⁴ - 8x³ + 20x² - 32x + 64