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 May 22nd, 2015, 04:47 AM #1 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 Roots of the complex equations Find all the roots for the following equation. $2x^4-x^3-x^2+3x+1=0$ My attempt, I factorised it to $(x+1)(2x^3-3x^2+2x+1)=0$ So I know one of its roots is -1. How to proceed then? Is there way to find the complex solution without using computer algebra system? May 22nd, 2015, 05:36 AM #2 Senior Member   Joined: Mar 2011 From: Chicago, IL Posts: 214 Thanks: 77 Cardano's method. May 22nd, 2015, 05:38 AM #3 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms I don't know how you're expected to find the roots. If you can find them numerically, use bisection between -1 and 0 and then divide out the approximate root; with what's left you can use the quadratic formula to get the complex roots. If you need a symbolic solution you need Cardano's formula. May 22nd, 2015, 06:31 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,967 Thanks: 2216 You need to solve $2x^3 − 3x^2 + 2x + 1 = 0$. Substituting $x = u + 1/2$ and dividing by 2 gives $u^3 + (1/4)u + 3/4 = 0$. Now let $3ab = -1/4$ and $a^3 + b^3 = 3/4$ and (carefully) solve these equations by obtaining and solving a quadratic equation whose roots are $a^3\!$ and $b^3\!$. It remains to solve $u^3 - 3abu + a^3 + b^3 = 0$, i.e., $(u + a + b)(u + \omega a + \omega^2b)(u + \omega^2 a + \omega b) = 0$, where $\omega$ is a complex cube root of 1, such as $-\frac12 + \frac{\sqrt3}{2}i$, and hence find $x$. Tags complex, equations, roots Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post WWRtelescoping Complex Analysis 6 January 27th, 2014 05:04 AM HugoFS Complex Analysis 8 July 25th, 2013 06:51 AM chocolatesheep Algebra 1 June 25th, 2013 03:27 PM surfcast23 Complex Analysis 7 January 27th, 2012 08:20 PM xboxlive89128 Complex Analysis 3 September 1st, 2009 06:38 PM

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