2016 is coming!

Algebra Level 4

x 1 2016 + x 2 2016 + + x 2016 2016 x_{1}^{2016} + x_{2} ^{2016} +\cdots+ x_{2016}^{2016}

If x 1 , x 2 , , x 2016 x_{1}, x_{2},\ldots, x_{2016} are the roots of the equation x 2016 2016 x 2 = 0 , x^{2016} - 2016x - 2 = 0, then find the value of the expression above.


The answer is 4032.

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4 solutions

Dev Sharma
Dec 28, 2015

x 2016 2016 x 2 = 0 x^{2016} - 2016x - 2 = 0

x 2016 = 2016 x + 2.... ( 1 ) x^{2016} = 2016x + 2 .... (1)

Putting x = x 1 , x 2 , . . . . . , x 2016 x = x_{1}, x_{2}, ....., x_{2016} in ( 1 ) (1) repeatedly, and add them, we get

x 1 2016 + x 2 2016 + . . . . . . . . + x 2016 2016 = 2016 ( x 1 + x 2 + . . . . . + x 2016 ) + 4032 x_{1}^{2016} + x_{2} ^{2016} + ........ + x_{2016}^{2016} = 2016(x_{1} + x_{2} + ..... + x_{2016}) + 4032

Using Vieta, x 1 + x 2 + . . . . + x 2016 = 0 x_{1} + x_{2} + .... + x_{2016} = 0

So the value of expression is x 1 2016 + x 2 2016 + . . . . . . . . + x 2016 2016 = 4032 x_{1}^{2016} + x_{2} ^{2016} + ........ + x_{2016}^{2016} = 4032

Akshat Sharda
Dec 29, 2015

x 2016 2016 x 2 = 0 x 2016 = 2016 x + 2 x 2016 = ( 2016 x + 2 ) = 2016 x + 2 = 2016 ( 0 ) + 2016 2 = 4032 \begin{aligned} x^{2016}-2016x-2 & = 0 \\ x^{2016} & = 2016x+2 \\ \sum x^{2016} & = \sum \left(2016x+2\right) \\ & = 2016 \sum x+ \sum 2 \\ & = 2016\left(0\right)+2016\cdot 2 \\ & = \boxed{4032} \end{aligned}

This solution is similar to mine.

Dev Sharma - 5 years, 5 months ago
Manuel Kahayon
Dec 29, 2015

Some other solution... By Newton's sums

x 1 2016 + x 2 2016 + + x 2016 2016 = S 2016 x^{2016}_1+ x^{2016}_2+ \ldots + x^{2016}_{2016} = S_{2016}

S 2016 + S 2015 a 1 + + S 1 a 2015 + 2016 a 2016 = 0 S_{2016} + S_{2015} \cdot a_{1} + \ldots + S_1 \cdot a_{2015} +2016 \cdot a_{2016} = 0

Since a 1 , a 2 , a 3 , . . . a 2014 = 0 a_1, a_2, a_3, ... a_{2014}=0 and S 1 = a 1 = 0 S_1 = -a_1 = 0 , then

S 2016 + S 2015 0 + + 0 a 2015 + 2016 a 2016 = 0 S_{2016} + S_{2015} \cdot 0 + \ldots + 0 \cdot a_{2015} +2016 \cdot a_{2016} = 0

S 2016 + 2016 a 2016 = 0 S_{2016} + 2016 \cdot a_{2016} = 0

S 2016 + 2016 ( 2 ) = 0 S_{2016} + 2016 \cdot (-2) = 0

S 2016 = 4032 S_{2016} = 4032

Rahul Saxena
Feb 25, 2016

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