Math help polynomials

Question

whats the simplest polynomial one can get from the zeros 8i, 3, and -3

Answer 1

Student,

 

For the problem as worded, David W. gave you the correct answer.  But perhaps you really meant to ask:

 

"What is the simplest polynomial WITH REAL COEFFICIENTS that one can get from the zeros 8i, 3, and -3?"

 

If that is the real question, then you need to recall that if you have a complex zero, its complex conjugate is also a zero.  The simplest polynomial will have the lowest possible degree for the set of zeroes, and a leading coefficient of 1.  The zeros are

 

8i

-8i

3

-3

 

The degree is 4.  The simplest polynomial is the expansion of

 

p(x) = (x - 8i) [x - (-8i)] (x - 3) [x - (-3)] = (x² + 64)(x² - 9) = ?

 

You can expand this further if you wish.

Answer 2

(x - 8i)(x - 3)(x + 3) = your polynomial