Question

What is sum of the imagenary part of the roots of a polynomial

Answer 1

Answer:

As worded, the answer is "whatever imaginary number you would like it to be".

However, presumably the question is meant to relate only to a polynomial with real coefficients.  In that case, the answer is 0.

Step-by-step explanation:

The roots of a polynomial with real coefficients turn up in conjugate pairs, and in each pair, the sum of the imaginary parts is 0 [note (a+bi) + (a-bi) = 2a; the imaginary parts cancel].  So the sum of the imaginary parts of all the roots is 0.

Another way you might think about it is this.

For any polynomial

$a_nx^n + a_{n-1}x^{n-1} +\cdots$

the sum of all the roots is given by

$-a_{n-1}/a_n$

So if $a_n$ and $a_{n-1}$ are real, then the sum of the roots is real, which means the imaginary part is 0.