$x_1 ,x_2, … , x_9$ are the roots of $x^9+7x-2$; then $(x_1)^9+(x_2)^9+ …+(x_9)^9=?$ [closed]

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.

closed as off-topic by C. Falcon, Daniel W. Farlow, Leucippus, JMP, NamasteJul 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.

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?

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 $$18$$

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