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Polunomials

I understand that X^6 - 22X^3 + 49 = 0 is the right polynomial for which (3 + 2^(1/2))^(2/3) is a solution. What I don't understand is how to get that polynomial because I keep getting x^3 - 6x^(2/3) + 7 = 0 and I know it is wrong. It just so happens that I know the correct polynomial is
X^6 - 22X^3 + 49 = 0 but I need to know how to get it given the solution
(3 + 2^(1/2))^(2/3). I must be making a simple alg. mistake hence I get
the polynomial x^3 - 6x^(2/3) + 7 = 0 which is clearly wrong. i.e. the given solution (3 + 2^(1/2))^(2/3) is not a solution for x^3 - 6x^(2/3) + 7 = 0. How do I go from the given solution to the correct polynomial which is X^6 - 22X^3 + 49 = 0?

here is may flawed work

This is what I am doing (3 + 2^(1/2))^(2/3)

x = (3 + 2^(1/2))^(2/3)

x^(3/2) = 3 + 2^(1/2)

x^(3/2) - 3 = 2^(1/2)

(x^(3/2) - 3)^2 = 2

(x^(3/2) - 3)^2 - 2 = 0 hence x^3 - 6x^(2/3) + 7 = 0 but I should get X^6 - 22X^3 + 49 = 0.

Solution Preview

Hi,

You are actually correct on the halfway part of the ...

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