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

[superelf83]

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...?

 

[TD]

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.

 

[HallsofIvy]

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.