Algebra #28/Trigonometry #6: My Prediction was Wrong
by djmathman, Nov 21, 2011, 2:29 AM
Problem wrote:
The polynomial
![\[ P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} \]](//latex.artofproblemsolving.com/9/9/0/99072bd65bf6cf99091db17698d62872a819ddc8.png)
has
complex roots of the form ![$ z_k = r_k[\cos(2\pi \alpha_k)+i\sin(2\pi \alpha_k)]$](//latex.artofproblemsolving.com/d/a/a/daafb3d828d1abd848cd2bce99b01a63c323a1d2.png)
,
with 
and 
Given that 
where 
and 
are relatively prime positive integers, find 
(SOURCE)
![\[ P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} \]](http://latex.artofproblemsolving.com/9/9/0/99072bd65bf6cf99091db17698d62872a819ddc8.png)
has
complex roots of the form ![$ z_k = r_k[\cos(2\pi \alpha_k)+i\sin(2\pi \alpha_k)]$](http://latex.artofproblemsolving.com/d/a/a/daafb3d828d1abd848cd2bce99b01a63c323a1d2.png)
,
with 
and 
Given that 
where 
and 
are relatively prime positive integers, find 
(SOURCE)
This polynomial, at first, seem very difficult to work with. We could try to multiply out the stuff in parenthesis, but doing so would be a pain. However, since the roots are in the form
![$r_k[\cos(2\pi \alpha _k)+i\sin(2\pi \alpha_k)]=r_ke^{2\pi i\alpha_k}$](http://latex.artofproblemsolving.com/3/8/b/38b18363ca46b4c98ab19bf06277cd03af43502a.png)
, we think of roots of unity. This suggests we write the stuff inside the parenthesis in a more concise manner:![\[P(x)=\left(\dfrac{x^{18}-1}{x-1}\right)^2-x^{17}\]](http://latex.artofproblemsolving.com/1/f/e/1fe5d0b465f12cdb27456ada05e27cfee869ee74.png)
Now we can expand and simplify:
Aha! We see that the first fraction deals with the 19th roots of unity besides
, and the second fraction deals with the 17th roots of unity besides . Since 
, we are on the right track.Now all that's left is to find the
smallest values of 
and add them together. Let's examine the first root: 
. This translates to ![$\cos \left(\dfrac{2\pi}{19}\right)+i\sin\left(\dfrac{2\pi}{19}\right)=\cos \left[2\pi\left(\dfrac{1}{19}\right)\right]+i\sin\left[2\pi\left(\dfrac{1}{19}\right)\right]$](http://latex.artofproblemsolving.com/b/4/0/b405a79f9716481897d8da5d40bb68f35dd95952.png)
, so 
. This means that all the roots are fractions with denominators 
or 
. The five smallest values of are 
, 
, 
, 
, and 
(a simple cross-multiply-to-compare check works here). Thus 
. Our answer is thus 
.This post has been edited 4 times. Last edited by djmathman, Apr 6, 2015, 2:55 AM
Reason: latex
Reason: latex
April 2025
November 2024