October 16th, 2018, 08:26 PM  #11  
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  Quote:
The very first root you found is not quite correct. It should be $z=3$ that's a root, and so $(z3)$ is a factor. Did you factor by grouping from the beginning? This will make things easier, but as Micrm@ss said, it's sort of a lucky problem.  
October 16th, 2018, 08:28 PM  #12  
Senior Member Joined: Nov 2015 From: United States of America Posts: 181 Thanks: 21 Math Focus: Calculus and Physics  Quote:
 
October 16th, 2018, 08:29 PM  #13  
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  Quote:
 
October 16th, 2018, 08:31 PM  #14 
Senior Member Joined: Nov 2015 From: United States of America Posts: 181 Thanks: 21 Math Focus: Calculus and Physics  
October 16th, 2018, 08:32 PM  #15 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  You're welcome. It's easy to make sign errors on these types of problems.
Last edited by ProofOfALifetime; October 16th, 2018 at 08:35 PM. 
October 16th, 2018, 08:45 PM  #16 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,780 Thanks: 970  
October 16th, 2018, 10:28 PM  #17 
Global Moderator Joined: Dec 2006 Posts: 20,089 Thanks: 1902 
As the number of terms is even, grouping in pairs should be tried. You made a sign error. The equation can be written as z^4(z  3)  16(z  3) = 0, i.e. (z^4  16)(z  3) = 0, which gives (z^2 + 4)(z + 2)(z  2)(z  3) = 0. Finally, using imaginary numbers, (z + 2i)(z  2i)(z + 2)(z  2)(z  3) = 0. 
October 17th, 2018, 10:21 AM  #18  
Senior Member Joined: Nov 2015 From: United States of America Posts: 181 Thanks: 21 Math Focus: Calculus and Physics  Quote:
So with $z^416 = $ $\Rightarrow$ $(z^2 4)(z^2+4)$ utilizing difference of two squares. I like this approach. A lot quicker than rational root theorem. From here on out, I'll always try grouping first when the terms are even. I'll save rational root theorem as a last resort.  
October 17th, 2018, 12:47 PM  #19  
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  Quote:
 

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