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

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To solve the equation x^3 + x^2 - x - 1 = 0, it is established that x = 1 is a solution, making (x - 1) a factor. By factoring out (x - 1), the remaining quadratic factor can be determined using methods such as Horner's rule or by equating coefficients. This involves setting up the equation (x - 1)(x^2 + Ax + B) and solving for A and B based on the coefficients of the original polynomial. The quadratic equation derived from this process can then be solved to find the other roots. The discussion highlights the complexity of the general cubic formula, which is not necessary for this specific problem.
superelf83
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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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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.
 
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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