Find x in x^3+x^2-x-1=0 Equation

  • Thread starter superelf83
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In summary, to find the value of x in the equation x^3+x^2-x-1=0, you can use the fact that x=1 is a solution and factor out (x-1) to determine the remaining quadratic factor. Another method is to use the expressions (x-1)(x^2+Ax+B) and (x^3+Ax^2+Bx-x^2-Ax-B), where the coefficients must be equal for all x. Finally, there is a general cubic formula, but it is complex.
  • #1
superelf83
5
0
simple "find the x" question

how do you solve for x in this equation?

x^3+x^2-x-1=0

i know one of them is 1. but the other one...?
 
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  • #2
If you know that x = 1 is a solution, then (x-1) is a factor of the polynomial. Factor it out and determine the remaining (quadratic) factor, e.g. using Horner's rule.
 
  • #3
Another way to get the quadratic that is left is to write:
(x- 1)(x2+ Ax+ B)= x3+ Ax2+ Bx- x2- Ax- B= x3+ (A- 1)x2+ (B-A)x- B= x3+ x2- x- 1. In order for those to be equal for all x, corresponding coefficients must be the same: A- 1= 1, B- A= -1, -B= -1.
Solve for A and B and then solve the quadratic equation.

There is a general "cubic" formula but it is very compliciated.
 

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