
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 31st, 2014, 01:39 PM  #1 
Senior Member Joined: Oct 2014 From: Complex Field Posts: 119 Thanks: 4  Complex polynomial and Vieta's formulas
Hello, I had a question in my homework that I managed to solve part A of it with one of the Vieta's formulas, but part B was wrong. I am almost sure that part B also uses one of the Vieta's formulas but I just don't know how to use it properly. The question is: P(z)=z^7 + a6*z^6 + a5*z^5 + a4*z^4 + a3*z^3 + a2*z^2 +a1*z + a0 provided information: z1=1+2i is the root of P(z),P'(z) and P''(z) ; a0=53 A) Real root of P(z) is _____ ? B) a6=? My answer to A was : from the information we can say that (1+2i) is root of multiplicity 3. and because it's a complex number also the conjugate is of multiplicity 3. and the only one left is a real (because of 7 roots) so I mark it as 't' here. So I used Vieta's formula for multiplying and got: (1+2i)^3*(12i)^3*t = (1)^7*(a0/a7) => // a0=53 ; a7=1 => 125*t = (1)*(53/1) =>t = 53/125 This is part A and is correct. Now what I tried to do for part B was to use the Vieta's sum formula in this way: The sum of complex roots is 0, and the remaining root is 53/125. So: 0+53/125 = a6/a7 => // a7=1 a6=53/125 < But that's a wrong answer. What am I doing wrong? Thanks! 
October 31st, 2014, 02:20 PM  #2  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
Also from what you have written it is evident that you think you are dealing with a real polynomial, but have nowhere stated it. CB  
October 31st, 2014, 02:33 PM  #3 
Senior Member Joined: Oct 2014 From: Complex Field Posts: 119 Thanks: 4 
Thanks, But as for part A, did I do something wrong and by chance the answer is correct? Or part A is alright? And also give me a tip for part B? Last edited by noobinmath; October 31st, 2014 at 02:36 PM. 
October 31st, 2014, 03:06 PM  #4  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
For part B the sum of the roots is 6+53/125 (assuming you answer to part A is numerically correct). This is because each of the complex roots has real part 1 and so the real part of the sum of roots is 6+53/125 and the imaginary parts cancel as the complex roots occur in conjugate pairs. CB  
October 31st, 2014, 03:12 PM  #5 
Senior Member Joined: Oct 2014 From: Complex Field Posts: 119 Thanks: 4 
Thank you, the first part was numerically correct. But can you tell me what is the real approach for it? Because I want to get to the right answer in the right way, and not by a chance. And also, how come that I did it in the real polynomial approach and still got the right answer? (How would you do that?)
Last edited by noobinmath; October 31st, 2014 at 03:19 PM. 
October 31st, 2014, 11:20 PM  #6  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
What we know is that the product of the roots is real the product of the four mystery roots is a real multiple of $(12i)^3$ and that is all we can say which is insufficient to fix even one real root. For example the following polynomial satisfies everything given but does not have real coefficients other than the first and last: $$ P(x)=\left( x1\right) \,{\left( x\frac{\sqrt{53}}{{5}^{\frac{3}{2}}}\right) }^{2}\,{\left( x2\,i1\right) }^{3}\,\left( x2\,i+11\right) $$ and it has a real root different from the "correct answer" What I am trying to say is that there is insufficient information to solve the asked questions A and B without the a's being real. CB Last edited by CaptainBlack; October 31st, 2014 at 11:23 PM.  
November 1st, 2014, 01:29 AM  #7 
Senior Member Joined: Oct 2014 From: Complex Field Posts: 119 Thanks: 4 
Oh so the polynomial is considered "real" not because of "z" but because of the a's? So yes, the a's are real!, sorry for forgetting to mention that. And now B part is also correct, thank you! (It is 6+53/125) 

Tags 
complex, formulas, polynomial, vieta 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Complex polynomial  muhsentdrawjiac  Algebra  5  July 12th, 2012 12:41 AM 
absolute value of a complex polynomial  Broodjekaas  Complex Analysis  1  June 9th, 2011 06:30 AM 
integral of a complex polynomial  Broodjekaas  Complex Analysis  0  June 8th, 2011 01:14 AM 
complex polynomial  Vygotsk  Complex Analysis  2  May 17th, 2010 03:31 PM 
complex polynomial  ^e^  Complex Analysis  1  April 1st, 2007 04:45 AM 