Complex zeros of polynomial functions

Question

How do you Write a polynomial function of these degrees with a real coefficients instead of form that has the given zeros The problem is I honestly don’t even know where to start -5,3,4+I

Answer 1

Given a set of zeros z₁, z₂, z₃, etc., start with

f(x) = A(x - z₁)(x - z₂)...(x - z_n)

where A is real and A≠ 0. To ensure the coefficients are real, the complex conjugate of any complex zero must also be a zero, so your minimum set of zeros is

-5

3

4 + i

4 - i

and the polynomial of minimum degree is

f(x) = A(x + 5)(x - 3)[x - (4 + i)][x - (4 - i)]

If you expand this, the coefficients will all come out real. You will need the value of the function at some x in order to find A. If you are told that the leading coefficient is 1, then A = 1.