If x = 2 - 2^{1/3} + 2^{2/3}, then the value of x^3-6x^2+18x+18 is:

Algebra and Applications Test 3

Question : 16 of 25

Marks: +1, -0

x = 2 - 2^{1/3} + 2^{2/3}

, then the value of

x^3-6x^2+18x+18

is:

Solution:

x = 2 - 2^{1/3} + 2^{2/3}

x - 2 = 2^{2/3} - 2^{1/3}

Cubing both sides we get:

(x - 2)^3 = (2)^2 - (2)^1 - 3 × 2 (2^{2/3} - 2^{1/3})

x^3 - 8 - 6x (x - 2) = 4 - 2 - 6 (x - 2)

x^3 - 8 - 6x^2 + 12x = 2 - 6x + 12

x^3 - 6x^2 + 18x - 40 + 18 = 0

x^3 - 6x^2 + 18x + 18 = 40

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