Question

How do you find all additional roots given the roots \[-4i \, \And \, 6-i\]?

Answer 1

Hint: To solve the given problem, we will assume the function has only four roots and they are conjugate pairs of each other. That is the other roots of the given function are conjugates of the given roots. Now, we will look at how to calculate the conjugate of a complex number. The complex conjugate of a complex number of the form \[a+ib\] is \[a-ib\],here a and b are any real numbers.

Complete step by step solution:
We are given the two roots as \[-4i \, \And \, 6-i\] and asked to find the additional roots. For this problem, we will assume that the additional roots and given roots are conjugate pairs of each other, so we can find the additional roots by simply taking conjugate of the given roots.
The first roots are \[-4i\], here we have a and b as 0 and 4 respectively. Hence, the conjugate of this complex number is \[-\left( -4i \right)=4i\]. The second root is \[6-i\], here we have a and b as 6 and -1 respectively. Hence, the conjugate of this complex number is \[6-\left( -i \right)=6+i\].

Thus, the additional roots are \[4i\] and \[6+i\].

Note: Here, we assumed that the additional roots and the given roots are conjugate pairs of each other, as it was the simplest case and we did not have any additional information about the function like its degree, etc. If we were given any more information, we would have to use it to solve the given question.