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If x + 1/x = sqrt(3), then find x^{18} + x^{12} + x^6 + 1.

 Jun 13, 2020
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x+1/x=√3 (Let us assume this one as a+b form)

a^2+b^2 = (a+b)^2–2ab ( apply this formula in above)

x^2+1/x^2=1

We know that

a^3+b^3= (a+b)(a^2-ab+b^2)

Now => x^3+1/x^3= 0

x^6=-1 ==> x^6+1=0

Coming to the question…

x^18+x^12+x^6+1

x^12(x^6+1)+1(x^6+1) = 0 (since above equation x^6+1=0)

Final answer=zero.

 Jun 13, 2020

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