Question

How do you form a polynomial function whose zeros are 3 and 4 + i?

Answer 1

Convert zeros to factors and multiply out to find the simplest possible polynomials.

Explanation:

If `a` is a zero of a polynomial `f(x)`, then `(x-a)` is a factor and vice versa.

So the simplest polynomial with these zeros is:

`(x-3)(x-(4+i)) = x^2-(7+i)x+(12+3i)`

If you want Real coefficients, then the Complex conjugate `4-i` must also be a zero and we find the simplest polynomial is:

`(x-3)(x-(4+i))(x-(4-i))`

`=(x-3)((x-4)^2-i^2)`

`=(x-3)(x^2-8x+17)`

`=x^3-11x^2+41x-51`

Any polynomial with these zeros must be a multiple (scalar or polynomial) of these 'simplest' polynomials.