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February 23rd, 2016, 12:22 PM  #1 
Member Joined: Dec 2015 From: Europe Posts: 42 Thanks: 1  Hard factorization problem
How can you factor $\displaystyle k^{10}+k^5+1$ to this beautiful form: $\displaystyle (k^2+k+1)(k^8k^7+k^5k^4+k^3k+1)$ Last edited by greg1313; February 23rd, 2016 at 12:48 PM. Reason: Corrected latex for k^10 
February 23rd, 2016, 04:36 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
$\displaystyle \begin{align*}k^{10} + k^5 + 1 &= \frac{k^{15}  1}{k^3  1}\cdot \frac{k^3  1}{k^5  1} \\ &= (k^{12} + k^9 + k^6 + k^3 + 1)\cdot \frac{k^2 + k + 1}{k^4 + k^3 + k^2 + k + 1} \\ &= (k^8  k^7 + k^5  k^4 + k^3  k + 1)(k^2 + k + 1)\end{align*}$ The longer factor = $\displaystyle (k^2  2\cos(24Â°)k + 1)(k^2  2\cos(48Â°)k + 1)(k^2  2\cos(96Â°)k + 1)(k^2  2\cos(192Â°)k + 1)$. 

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