If x+ \frac{1}{x}=\sqrt{3} then the value of x^{18}+x^{12}+x^{6}+1 is

SSC CPO SI and ASI 2011 Paper

Question : 139 of 200

Marks: +1, -0

x+ {1}/{x}=√{3}

then the value of

x^{18}+x^{12}+x^{6}+1

Solution:

(1)

x+ {1}/{x}=√{3}

⇒(x+ {1}/{x} )^{3}=x^{3}+ {1}/{x^{3}}+3 (x+ {1}/{x} )

(√{3})^{3}=x^{3}+ {1}/{x^{3}}+3(√{3})

⇒ x^{3}+ {1}/{x^{3}}=0

Now,

x^{18}+x^{12}+x^{6}+1

=x^{12} (x^{6}+1 )+1 (x^{6}+1 )

x^{12} (x^{6}+1 )+1 (x^{6}+1 )

= (x^{12}+1 ) (x^{6}+1 )

(x^{12}+1 ) x^{3} (x^{3}+ {1}/{x^{3}} )=0

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