Find a third degree polynomial function with real coefficients and with zeros 4 and 3+i

X = 4; x = 3 + i; x = 3 - i

(If you get a zero that is adding or subtracting, you always need to write it twice but change the sign do they cancel out)

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

Distributing the last two parenthesis first is always the best way to start off

(x-3-i) (x-3+i) has (x-3) in common so it can be separated to

(x-3) ^2 + (-i) (+i)

(x^2 - 6x + 9); (-i) (+i) is always + 1

(x^2 - 6x + 9) + 1

(x^2 - 6x + 10)

Now multiply this with (x-4)

x^3 - 6x^2 + 10x

- 4x^2 + 24x - 40

x^3 - 10x^2 + 34x - 40 = f (x)