Question

How do you find all additional roots given the roots `-4i` and `6-i`?

Answer 1

If these are roots of a polynomial equation with Real coefficients, then their conjugates, `4i` and `6+i` will also be roots.

Explanation:

If a polynomial equation has Real coefficients, then any Complex roots will occur in Complex conjugate pairs.

The Complex conjugate of `a+bi` is `a-bi`

So in our example, `4i` and `6+i` would be roots of any polynomial equation with Real coefficients that has `-4i` and `6-i` as roots.

The same is true of many equations involving functions with Real coefficients too, but breaks down in some cases involving `n`th roots, since we have conventions like `sqrt(-1) = i` rather than `sqrt(-1) = -i`. The simplest such example would be:

`x = sqrt(-1)`

Which has only Real coefficients and root `x = i`, but `x = -i` is not a root.