Question

if x=2+(2^1/3)+(2^2/3),then find the value of x³-6x²+6x-2

Answer 1

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

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

/* Cubing on both sides , we get */

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

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

/* By Algebraic Identity */

$\pink { (a-b)^{3} = a^{3} - 3a^{2}b+3ab^{2} - b^{3} }$

$\orange { (a+b)^{3} = a^{3} + b^{3} +3ab(a+b) }$

$\implies x^{3} - 6x^{2} + 12x - 8 \\= 2 + 4 + 3\times 2^{\frac{1+2}{3}} ( x - 2 ) \: [ From \:(1) ]$

$\implies x^{3} - 6x^{2} + 12x - 8 \\= 6 + 3\times 2^{\frac{3}{3}} ( x - 2 )$

$\implies x^{3} - 6x^{2} + 12x - 8 \\= 6 +6 ( x - 2 )$

$\implies x^{3} - 6x^{2} + 12x - 8 \\= 6 + 6x -12$

$\implies x^{3} - 6x^{2} + 12x - 8 -6 - 6x +12= 0$

$\implies x^{3} - 6x^{2} + 6x - 2= 0$

Therefore.,

$\red { Value \: of \: x^{3} - 6x^{2} + 6x - 2}\green {= 0}$

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