Solve the equation #x^4+4x^3+8=0#?
Obtain the resolvent cubics, by Descartes' method and by Ferrari's method, of the equation #x^4+4x^3+8=0# . Are the cubics the same? Further, Use either method to obtain the roots of this equation.
Obtain the resolvent cubics, by Descartes' method and by Ferrari's method, of the equation
1 Answer
See below.
Explanation:
Consider the polynomial
and also the expansion
(here
now substituting
and now choosing properly
or
///////////////////////////////////////////////////////////////////////////////////////////
Note that solving
////////////////////////////////////////////////////////////////////////////////////////////
Now solving for
Now
and the former quartic remains
resulting in the resolution of
This is left as an exercise for the proficient reader.
NOTE: This is the so called Ferrari's method.