Complex zeros of polynomials

Question

Find a degree 3 polynomial with real coefficients having zeros 3and 4i and a lead coefficient of 1. Write P in expanded form.

Answer 1

Since 4i is a root, so is -4i. Also, if c is a root, then x - c is a factor..

P(x) = (x - 3)(x - 4i)(x+4i) = (x-3)(x² + 16) = x³ - 3x² + 16x - 48

Answer 2

P(x) = (x-3)( (x - 3) + 4i) ( (x-3) - 4i)

= (x-3) [ (x-3)^2 - (4i)^2]

= (x-3) [ x^2 - 6x + 9 + 16 ]

= (x-3)(x^2 - 6x + 25)

= x^3 - 6x^2 + 25x

-3x^2 + 18x - 75

= x^3 - 9x^2 + 43x - 75