Form a polynomial f(x)
posted by Hailey on .
Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree 5;
Zeros: 3; i; 6+i
F(x)=a ( )

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  (6i)) 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.