Question

If `x=1+i` is a factor of `ax^2+bx+c` then what is another factor?

Answer 1

c) `1-i`

Explanation:

I will assume that by 'factor' you intended 'zero' or 'root'.

A 'factor' would be something like `(x - 1 - i)`, which corresponds to a 'zero' `x=1+i` of `ax^2+bx+c`, also called a 'root' of `ax^2+bx+c = 0`.

If `x = 1+i` is a zero of `ax^2+bx+c` and `a`, `b` and `c` are Real numbers then `x = 1-i` is the other zero.

If `a`, `b` and `c` can be Complex numbers, then the other zero can be anything.

In general, given any polynomial with Real coefficients, any zeros are Real or occur in Complex conjugate pairs.

From the perspective of the Real numbers, there is no way to tell the difference between `i` and `-i`. The only thing the Real numbers 'know' about `+-i` is that they are square roots of `-1`.