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  4. If x=2+22 / 3+21 / 3, then find the value of x3-6 x2+6 x

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Q. If $x=2+2^{2 / 3}+2^{1 / 3}$, then find the value of $x^{3}-6 x^{2}+6 x$

Complex Numbers and Quadratic Equations

Solution:

$x=2+2^{2 / 3}+2^{1 / 3} $
$ x^{3}-6 x^{2}+6 x$
$x-2=\left(2^{2 / 3}+2^{1 / 3}\right)\,\,\,...(i)$
making whole cube
$(x-2)^{3}=\left(2^{2 / 3}+2^{1 / 3}\right)^{3}$
$x^{3}-8-6 x(x-2)$
$=4+2+3 \cdot\left[2^{2 / 3} \cdot 2^{1 / 3}\right]$
$\left(2^{2 / 3}+2^{1 / 3}\right)$
$x^{3}-8-6 x^{2}+12 x$
$=6+6(x-2)$
$x^{3}-8-6 x^{2}+12 x$
$=6+6 x-12$
$x^{3}-6 x^{2}+6 x=2$