First slide

Theory of equations

Question

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

Easy

A

3
B

2
C

1
D

None of these

Solution

$Given x - 2 = 2^{2 / 3} + 2^{1 / 3}$
cubing on both sides
$(x - 2)^{3} = 2^{2} + 2 + 3 \cdot 2^{2 / 3} \cdot 2^{1 / 3} (2^{2 / 3} + 2^{1 / 3})$
$(x - 2)^{3} = 2^{2} + 2 + 3 \cdot 2^{2 / 3} \cdot 2^{1 / 3} (x - 2) = 6 + 6 (x - 2)$
$\begin{array}{l} or x^{3} - 6 x^{2} + 12 x - 8 = - 6 + 6 x \\ ∴ x^{3} - 6 x^{2} + 6 x = 2 \end{array}$

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