Question

form a polynomial f(x) with real coefficients having the given degree and zeros. degree 5; zeros -7; -i;-9+i

enter the polynomial.

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 -9+i, there will also be -9 - i

so we know we have factors of (x+7) , (x^2 + 1) and two more

I will use the sum and product rule to find the other
sum of -9+i and -9 - i = -18
product of the above is 81 - i^2 = 81 + 1 = 82
resulting in the quadratic factor
x^2 + 18x + 82

so f(x) = (x+7)(x^2 + 1)(x^2 + 18x + 82)

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