Write a polynomial function with real coefficients that has an imaginary zero and an irrational zero.

Answer 1

Suppose you know i and √2 are zeros of a polynomial. You know the complex conjugate of a complex zero must also be a zero, and you know square roots come in pairs of ±. So you will have the zeros

i

-i

√2

-√2

With minimal degree and a leading coefficient of 1, an example of the polynomial is

f(x) = (x - i)(x + i)(x - √2)(x + √2) = (x² + 1)(x² - 2)