Explanation:
with 4 zeroes it means we are taking about a polynomial of 4th grade (the max. variable exponent is 4).
we can create the polynomial via factorization.
remember, a "zero" means that the polynomial has an overall value of 0, when x is that value.
when is a product of factors 0 ? when at least one of the factors is 0, because 0×anything = 0.
what is a factor of the polynomial that is 0, when x = n ?
(x - n)
when x = n, then this turns 0 and turns the whole remaining polynomial to 0 too.
so, we create for the given zeros :
y = (x - -3)(x - 3)x(x - 4i) = (x + 3)(x - 3)x(x - 4i) =
= (x² - 9)x(x - 4i) = (x³ - 9x)(x - 4i) =
= x⁴ - 4ix³ - 9x² + 36ix = x⁴ - 9x² - 4i(x³ - 9x)