If #x = 5 - 5 ^ ( 2/3) - 5 ^ (1/3)#, prove that # x ^3 - 15x^2 + 60 x - 20 = 0#?

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Answer 1 · 2018-05-08T11:16:33

Please see a Proof in the Explanation.

#x=5-5^(2/3)-5^(1/3)#.

#:. 5-x=(5^(2/3)+5^(1/3))............(ast)#.

#:. (5-x)^3=(5^(2/3)+5^(1/3))^3#.

#:. 5^3-x^3-3(5)(x)(5-x)=(5^(2/3))^3+(5^(1/3))^3#

#+3xx5^(2/3)xx5^(1/3)(5^(2/3)+5^(1/3)), i.e., #

#125-x^3-75x+15x^2=5^2+5^1+3xx5xx(5-x)...[because, (ast)]#

#:. 125-75x+15x^2-x^3=30+75-15x#.

#:. 20-60x+15x^2-x^3=0,# as desired!