#"First put y =1/x, then we have"#
#341 = (1 - y^5)/(1 - y) = 1+y+y^2+y^3+y^4#
#=> y^4+y^3+y^2+y-340 = 0#
#"Rational roots theorem : search for divisors of 340 as"#
#"possible rational roots, we find y=4."#
#"We divide away the factor (y-4) as such : "#
#=>(y-4)(y^3+5y^2+21y+85) = 0#
#"The remaining cubic equation does not have rational roots"#
#"though, we can solve it with the Vieta substitution :"#
#"With the substitution y = z-5/3 we get"#
#z^3 + (38/3) z + (1600/27) = 0#
#"Substituting z=q p in z³ + b z + c = 0, yields :"#
#p^3 + b p / q^2 + c / q^3 = 0#
#"if we take q = sqrt(|b|/3), the coefficient of z becomes 3 or -3,"# #"and we get :"#
#"(here q = 2.05480467)"#
#p^3 + 3 p + 6.83037563 = 0#
#"Substituting p = t - 1/t, yields :"#
#t^3 - 1/t^3 + 6.83037563 = 0#
#"Substituting u = t³, yields the quadratic equation :"#
#u^2 + 6.83037563 u - 1 = 0#
#"A root of this quadratic equation is u=0.14339446."#
#"Substituting the variables back, yields :"#
#t = "cuberoot(u)" = 0.52341254.#
#p = -1.38712632.#
#z = -2.85027363.#
#y = -4.51694030.#
#"The other roots can be found by dividing and solving the"# #"remaining quadratic equation."#
#"They are complex : -0.24152985 "pm" 4.33124826 i."#
#"So x = 1/y are the solutions"#
#"so we have 1/4 and -0.2213888 as real solutions."#
