This problem is as easy as abc

Algebra Level 3

Given that x x is a complex number satisfying x + 1 x = 3 x + \dfrac{1}{x} = 3 , find the value of x 5 + 1 x 5 x^5 + \dfrac{1}{x^5} .


The answer is 123.

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Hobart Pao by Hobart Pao , 23, USA

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1 solution

Rishabh Jain
Jan 25, 2016

x 5 + 1 x 5 = ( x 2 + 1 x 2 ) ( x 3 + 1 x 3 ) − ( x + 1 x ) x^5 + \dfrac{1}{x^5}=(x^2 + \dfrac{1}{x^2})(x^3+ \dfrac{1}{x^3})-(\color{#D61F06}{x+ \dfrac{1}{x}}) = ( ( x + 1 x ) 2 − 2 ) ( ( x + 1 x ) 3 − 3 ( x + 1 x ) ) − ( x + 1 x ) =((\color{#D61F06}{x + \dfrac{1}{x}})^2-2)((\color{#D61F06}{x+ \dfrac{1}{x}})^3-3(\color{#D61F06}{x+ \dfrac{1}{x}}))-(\color{#D61F06}{x+ \dfrac{1}{x}}) = ( ( 3 ) 2 − 2 ) ( ( 3 ) 3 − 3 ( 3 ) ) − ( 3 ) =( (\color{#D61F06}{3})^2-2)((\color{#D61F06}{3})^3-3(\color{#D61F06}{3}))-(\color{#D61F06}{3}) = ( 7 ) ( 18 ) − ( 3 ) = 123 =(7)(18)-(3)=\Large\boxed{\boxed{123}}

9 Helpful 0 Interesting 1 Brilliant 0 Confused

Nice Solution...See this ...how to simplify x 5 + 1 x 5 . x^5+\dfrac{1}{x^5}.

A Former Brilliant Member - 5 years, 4 months ago

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Unfortunately I have not learnt about Vieta's yet ... :( BTW Thanks for the link. I'll read that as soon as I'll get time.

Rishabh Jain - 5 years, 4 months ago

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Welcome...This can help u in simplying any powers.. :)

A Former Brilliant Member - 5 years, 4 months ago

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