How can I find the roots of x 3 − 3 x + 1 using Cardano's formula?So far, I found that x = − 1 + − 3 2 3 + − 1 − − 3 2 3 x = − 1 + i 3 2 3 + − 1 − i 3 2 3 x = − 1 2 + i − 3 2 3 + − 1 2 − i − 3 2 3 I am now trying to express each cubic radicand in their exponential form.

Question

How can I find the roots of       x    3    −  3  x  +  1 using Cardano&#039;s formula?So far, I found that  x  =                              −          1          +                      −            3                          2              3    +                              −          1          −                      −            3                          2              3    x  =                              −          1          +          i                      3                          2              3    +                              −          1          −          i                      3                          2              3    x  =                              −          1                2            +                        i                      −            3                          2              3    +                              −          1                2            −                        i                      −            3                          2              3  I am now trying to express each cubic radicand in their exponential form.

kriteria0b1 · Accepted Answer

Use the Cardano approach:Let  x  =  u  +  vthen we will get      u    3    +      v    3    +  3  (  u  v  −  1  )  (  u  +  v  )  +  1  =  0by setting   u  v  −  1  =  0 we get  u  v  =  1and      u    3    +      v    3    =  −  1by cubing the first equation we obtain a simple quadratic system in terms of       u    3   and       v    3   which you can be reduced to a quadratic equation in terms of       u    3   for example:      u    3    +      1          u      3        +  1  =  0Then use the Quadratic formula to find       u    3   and       v    3    =  −  1  −      u    3  After extracting cubic roots of the obtained solution you need to take into account that uv is real. This will give the appropriate pairs of u and v whose sums will yield all three roots of the equation.

How can I find the roots of x^3-3x+1 using Cardano's formula?

Answered question

Answer & Explanation

New Questions in Algebra II