Question

If x = 2 + 2^(2/3) + 2^(1/3). Prove x^3 - 6x^2 + 6x -2 = 0

Answer 1

Given, x =  $2+2^{\frac{2}{3}} + 2^{\frac{1}{3}}$

= $(x-2) = 2^{\frac{2}{3}} + 2^{\frac{1}{3}}$

On cubing both sides, we get,

$(x-2)^{3} = 2^{2} + 2+3$ × $2^{\frac{2}{3}}$ × $2^{\frac{1}{3}}$ × $( 2^{\frac{2}{3}} + 2^{\frac{1}{3}})$

$x^{3} -6x^{2} +12x-8 = 4+2+3$ × 2 × (x - 2)

$x^{3} -6x^{2} +6x = 2.$

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