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

Dec 26, 2016

If these are roots of a polynomial equation with Real coefficients, then their conjugates, $4 i$ 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 $a + b i$ is $a - b i$

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