Step-by-step explanation:
We are finding a polynomial of degree 3.
Since 4 and 5i are the zeroes of the polynomial,
the conjugate -5i must also be such a zero.
=> f(x) = a(x - 4)(x - 5i)(x + 5i)
=> f(x) = a(x - 4)(x² - 25i²)
=> f(x) = a(x - 4)(x² + 25)
When x = 1, f(1) = -234.
=> a(1 - 4)(1² + 25) = -234
=> a(-3)(-26) = -234
=> -78a = -234
=> a = 3
Hence,
f(x) = 3(x - 4)(x² + 25) = 3x³ - 12x² + 75x - 300.
