November 20th, 2016, 07:06 AM  #1 
Newbie Joined: Nov 2016 From: Brazil Posts: 2 Thanks: 0  Cubic function
It is known that (x1) is the root of the function f(x) = x³  x² + x + a. Find, if possible, all the Roots.
Last edited by GabrielT; November 20th, 2016 at 07:10 AM. 
November 20th, 2016, 07:21 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,440 Thanks: 1312 
That wording doesn't make sense. You probably meant that (x  1) is a factor of the polynomial, which would imply that a = 1. x³  x² + x  1 = (x  1)(x² + 1) = (x  1)(x + i)(x  i) 
November 20th, 2016, 07:37 AM  #3 
Senior Member Joined: May 2016 From: USA Posts: 673 Thanks: 283 
I think you mean that 1 is a zero of the function $x^3  x^2 + x + a.$ If that is correct, then $\exists\ p,\ q,\ r\ such\ that\ (x  1)(px^2 + qx + r) = x^3  x^2 + x + a.$ Do you know why? $\therefore px^2 + qx + r = \dfrac{x^3  x^2 + x + a}{x  1}.$ Finish it up. EFIT: Skipjack has a MUCH more direct method, but my method can be applied more generally. Last edited by JeffM1; November 20th, 2016 at 07:40 AM. 
November 20th, 2016, 09:59 AM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 2,568 Thanks: 1271 
synthetic division ... Code: 1] 1 1 1 a 1 0 1  1 0 1 (a+1) $x^3  x^2 + x  1 = (x1)(x^2 + 0x + 1)$ 

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