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If x=(5-sqrt(21))/2 then prove that (x^3...

If `x=(5-sqrt(21))/2` then prove that `(x^3+1/x^3)-5(x^2+1/x^2)+(x+1/x)=0`

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Given`,x= (5- sqrt(21))/(2)`
`therefore (1)/(x) = (2)/(5-sqrt(21)) = (2)/((sqrt(5) - sqrt(21))) xx((5 +sqrt(21)))/((5+ sqrt(21))) = (2(5+sqrt(21)))/((5)^(2) - (sqrt(21))^(2)) `
`therefore x +(1)/(x) = ((5 - sqrt(21))/(2))+((5+sqrt(21))/(2)) = (5- sqrt(21) + 5+ sqrt(21))/(2) = (10)/(2) = 5`
` therefore (x+(1)/(x))^(2) = 5^(2) = 25`
`rArr x^(2) +(1)/(x^(2)) + 2 xx x xx (1)/(x) = 25`
`rArr (x^(2) + (1)/(x^(2))) + 2 = 25 rArr (x^(2) +(1)/(x^(2))) = 25 - 2 = 23`
And,`x +(1)/(x) =5rArr (x^(2) +(1)/(x^(3))) = (5)^(3) = 125`
`x^(3) +(1)/(x^(2)) + 3xx x xx (1)/(x) (x +(1)/(x)) = 125`
`rArr (x^(3) +(1)/(x^(3))) + 3 xx 5 = 125`
`rArr(x^(3) +(1)/(x^(3))) = 125 -15 = 110`
`therefore (x^(3) +(1)/(x^(3))) - 5(x^(2) +(1)/(x^(2))) + (x + (1)/(x))`
` = 110- 5 xx 23 + 5 = 110 - 115 + 5 = 115 - 115 = 0`
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