Question

Form a polynomial f(x) with real coefficients having the given degree and zeros.

Degree 5;
Zeros: -3; -i; -6+i
F(x)=a ( )

Answer 1

Complex numbers always appear as conjugate pairs, so if you have -i, then you also have +i
and if you have -6+i, there will also be -6 - i

so we know we have factors of (x+3) , (x^2 + 1) and two more

I will use the sum and product rule to find the other
sum of -6+i and -6 - i = -12
product of the above is 36 - i^2 = 36 + 1 = 37
resulting in the quadratic factor
x^2 + 12x + 37
You could also expand (x -(-6+i))(x - (-6-i)) and get the same result

so f(x) = (x+3)(x^2 + 1)(x^2 + 12x + 37)

notice, if expanded this will give you a 5th degree polynomial. If you have to expand it, do it very carefully and patiently.