Let's say I have the equation 2x^4 + 4x^3 -72x^2 + 24x and I want to factor it. Obviously one of the roots is zero, but the other 3 roots are all irrational. Is there a way to factor for irrational roots? If so could you explain how to factor for those roots?
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You've heard of the quadratic formula I expect... well, there are also similar (but increasingly more complicated) formulas for cubics and quartics as well. Most people don't bother memorizing them and are satisfied just knowing that such a formula exists that they could look up if they ever need it. Importantly, you should also be made aware that no such formula exists for a quintic or above... not because such a formula hasn't been found yet but rather that it has been proven that no such generic formula could exist that works for all cases using only radicals. – JMoravitz Mar 29 '23 at 13:35
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See General cubic formula, see General quartic formula, and finally see Abel-Ruffini theorem for unsolvability of the quintic. Additionally, I recommend this youtube video by Mathologer for history and a derivation on the cubic formula that is easy to follow and entertaining. – JMoravitz Mar 29 '23 at 13:35
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An older related thread. I happened to answer that, so I won't jump to try and use it as duplicate target. For one, it is specific to cubics (which happens to be enough for your particular example). – Jyrki Lahtonen Mar 29 '23 at 13:39