if x²+x+1=0, then find the value of [x³+1/x³]³
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x²+x+1 = 0
⇒ x²+1 = -x
⇒ (x²+1)/x = -1
Now,
![[ x^{3}+ \frac{1}{ x^{3} }] ^{3} = [ (x+ \frac{1}{x} )^{3}-3.(x).(1/x)(x+1/x) ]^{3} \\ \\ or, (x+ \frac{1}{x} )^{3} [( x+ \frac{1}{x}) ^{2} -3]^{3}](https://tex.z-dn.net/?f=+%5B+x%5E%7B3%7D%2B+%5Cfrac%7B1%7D%7B+x%5E%7B3%7D+%7D%5D++%5E%7B3%7D+%3D+%5B+%28x%2B+%5Cfrac%7B1%7D%7Bx%7D+%29%5E%7B3%7D-3.%28x%29.%281%2Fx%29%28x%2B1%2Fx%29++%5D%5E%7B3%7D++%5C%5C++%5C%5C+or%2C++%28x%2B+%5Cfrac%7B1%7D%7Bx%7D+%29%5E%7B3%7D+%5B%28+x%2B+%5Cfrac%7B1%7D%7Bx%7D%29+%5E%7B2%7D+-3%5D%5E%7B3%7D+ "[ x^{3}+ \frac{1}{ x^{3} }] ^{3} = [ (x+ \frac{1}{x} )^{3}-3.(x).(1/x)(x+1/x) ]^{3} \\ \\ or, (x+ \frac{1}{x} )^{3} [( x+ \frac{1}{x}) ^{2} -3]^{3}")
![or, (\frac{ x^{2} +1}{x}) ^{3} [( \frac{ x^{2} +1}{x} )^{2}-3] ^{3}](https://tex.z-dn.net/?f=or%2C+++%28%5Cfrac%7B+x%5E%7B2%7D+%2B1%7D%7Bx%7D%29+%5E%7B3%7D+%5B%28+%5Cfrac%7B+x%5E%7B2%7D+%2B1%7D%7Bx%7D+%29%5E%7B2%7D-3%5D+%5E%7B3%7D++ "or, (\frac{ x^{2} +1}{x}) ^{3} [( \frac{ x^{2} +1}{x} )^{2}-3] ^{3}")
Put the value of (x²+1)/x from Equn (1) ,
=(-1)³ [(-1)²-3]³
= -(-2)³ = 8
⇒ x²+1 = -x
⇒ (x²+1)/x = -1
Now,
Put the value of (x²+1)/x from Equn (1) ,
=(-1)³ [(-1)²-3]³
= -(-2)³ = 8
karthik4297:
did you get answer?
thank u very much karthik. I have posted one more question please help me out with that too.
thanks
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