Best ways in solving cubic equations without information on roots

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SUMMARY

The discussion focuses on solving cubic equations without prior knowledge of their roots. A user successfully found one root, x=7, and reduced the cubic equation to a quadratic to find the remaining roots. Another participant suggests using the Rational Root Theorem as a more efficient method to identify potential rational roots before applying other techniques. This approach simplifies the process and often leads to discovering small integer roots quickly.

PREREQUISITES
  • Understanding of cubic equations and their properties
  • Familiarity with the Rational Root Theorem
  • Basic knowledge of quadratic equations
  • Experience with polynomial factorization techniques
NEXT STEPS
  • Study the Rational Root Theorem in detail
  • Learn methods for factoring cubic equations
  • Explore numerical methods for root-finding, such as Newton's method
  • Investigate synthetic division as a technique for simplifying polynomial equations
USEFUL FOR

Students learning algebra, mathematicians interested in polynomial equations, and educators seeking effective teaching methods for solving cubic equations.

cheahchungyin
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Homework Statement



Learning to solve cubic equations without the knowledge of any roots, and the easiest way I found out so far is still time-consuming

Homework Equations



The equation and attempt is shown in the image below, tell me if its unclear :shy:

The Attempt at a Solution



http://sphotos-a.ak.fbcdn.net/hphotos-ak-prn1/c67.0.403.403/p403x403/60637_450846098287003_1194154338_n.jpg

eventually found out x=7. Then by reducing the equation into a quadratic, I can find the other roots

So are there better ways in solving cubic? maybe some hacking fast ones? :D Appreciate the help! :smile:
 
Last edited by a moderator:
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You should try to find a rational root first using the Rational Root theorem. If you can find one, then you are practically done. Only if there aren't any would I resort to your technique.
 
In fact, doing that, it is easy to find three small integer roots.
 

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