Power to the roots

Algebra Level 5

Find the sum of the 17th powers of the 17 roots of the equation:

x 17 − 3 x + 1 = 0. x^{17}-3x+1=0.


The answer is -17.

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William Isoroku by William Isoroku , 25, USA

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1 solution

Sanjeet Raria
Nov 18, 2014

We've x 17 − 3 x + 1 = 0 x^{17}-3x+1=0 ⇒ x 17 = 3 x − 1 . . ( 1 ) \Rightarrow x^{17}=3x-1\space \space..(1) Let x 1 , x 2 , x 3 , . . . , x 17 x_1, x_2, x_3,...,x_{17} are roots of this equation. Putting x = x 1 , x 2 , x 3 , . . . , x 17 x= x_1, x_2, x_3,...,x_{17} repeatedly in ( 1 ) (1) and adding them. We get x 1 17 + x 2 17 + . . . + x 17 17 = 3 ( x 1 + x 2 + x 3 + . . . + x 17 ) − 17 {x_1}^{17}+ {x_2}^{17}+...+ {x_{17}}^{17}=3( x_1+x_2+ x_3+...+x_{17})-17 By Vieta, x 1 + x 2 + x 3 + . . . + x 17 = 0 x_1+x_2+ x_3+...+x_{17}=0 Hence x 1 17 + x 2 17 + . . . + x 17 17 = − 17 {x_1}^{17}+ {x_2}^{17}+...+ {x_{17}}^{17}=\boxed{-17}

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We can also use Newton's Sum for a straightforward solution:

P 17 + 0 P 16 + 0 P 15 … 0 P 2 − 3 P 1 + 17 ( 1 ) = 0 P_{17}+0P_{16}+0P_{15}\dots 0P_{2} -3P_1+17(1)=0

Christopher Boo - 6 years, 3 months ago

Did the same way , nice solution sir.

Shubhendra Singh - 6 years, 3 months ago

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