SOLUTION: Find a polynomial function with the integer coefficients that has the given zeros: 1+(sqrt)3i, 2, 2, -1-(sqrt)2 Thank you

Algebra ->  Trigonometry-basics -> SOLUTION: Find a polynomial function with the integer coefficients that has the given zeros: 1+(sqrt)3i, 2, 2, -1-(sqrt)2 Thank you      Log On


   

Question 986624: Find a polynomial function with the integer coefficients that has the given zeros:
1+(sqrt)3i, 2, 2, -1-(sqrt)2
Thank you
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Since 2 is written twice, I assume you mean the polynomial is to have
2 twice as a zero, or of multiplicity 2.

Conjugates of surds and complex numbers are also zeros:

x=1%2Bsqrt%283%29i, x=1-sqrt%283%29i, x=2, x=2, x=-1-sqrt%282%29, x=-1%2Bsqrt%282%29

x-1-sqrt%283%29i=0, x-1%2Bsqrt%283%29i=0, x-2=0, x-2=0, x%2B1%2Bsqrt%282%29=0, x%2B1-sqrt%282%29=0



Multiply all that out.  Takes a long time:

x%5E6-4x%5E5%2B3x%5E4%2B14x%5E3-48x%5E2%2B56x-16=0

So polynomial that has those zeros is

f%28x%29=x%5E6-4x%5E5%2B3x%5E4%2B14x%5E3-48x%5E2%2B56x-16

Edwin