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Let $x_1 ,x_2, ... , x_9$ are the roots of $x^9+7x-2$ then $(x_1)^9+(x_2)^9+ ...+(x_9)^9=?$

I have not figure it out yet, thanks in advance.

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closed as off-topic by C. Falcon, Daniel W. Farlow, Leucippus, JMP, Namaste Jul 1 '17 at 20:02

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Hint: Since $x_i^9 = -7x_i+2$, you just need to find the sum of the roots.

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Taking $x_1$ as an example

$$ (x_1)^9 = -7x_1+2 $$

so in general $$ (x_i)^9 = -7x_i+2 $$

So in the end you get:

$$ x_1^9 + ...+x_9^9 = -7(x_1+x_2+x_3+...+x_9)+18 $$

Can you proceed from here? What do you know about the sum of the roots of a polynomial?

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I plugged this polynomial into MATLAB using

P = roots[1, 0, 0, 0, 0, 0, 0, 0, 7, -2] 

and got

$$\left(\begin{array}{ccc|r} -1.2104 + 0.4893i \\ -1.2104 - 0.4893i \\ -0.5216 + 1.1818i \\ -0.5216 - 1.1818i \\ 0.4514 + 1.1829i \\ 0.4514 - 1.1829i \\ 1.1378 + 0.4906i \\ 1.1378 - 0.4906i \\ 0.2857 + 0.0000i \\ \end{array}\right)$$

Taking each term to the 9th power,

X = P.^9

you get

$$\left(\begin{array}{ccc|r} 10.4726 - 3.4253i \\ 10.4726 + 3.4253i \\ 5.6514 - 8.2728i \\ 5.6514 + 8.2728i \\ -1.1597 - 8.2805i \\ -1.1597 + 8.2805i \\ -5.9643 - 3.4343i \\ -5.9643 + 3.4343i \\ 0 \\ \end{array}\right)$$

The sum of all these terms is $$18$$

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  • $\begingroup$ And without a computer, how would you have done it? $\endgroup$ – José Carlos Santos Jun 30 '17 at 6:56
  • $\begingroup$ Nope...I'm too stupid. $\endgroup$ – Andy Jun 30 '17 at 20:51

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