Can I use a calculator? Please?

Algebra Level 3

Let $r$ and $s$ be the roots of the equation $x^2+6x-1=0$ . What is the value of $r^5+s^5$ ?

The answer is -8886.

1 solution

Chew-Seong Cheong
Jan 14, 2018

By Vieta's formula , we have $S_1 = r+s = -6$ and $S_2 =rs = -1$ . Using Newton's sums method and let $P_n = r^n + s^n$ , where $n$ is a positive integer.

$\begin{array} {} P_1 = r+s = S_1 & = -6 \\ P_2 = r^2+s^2 = S_1P_1 - 2S_2 = (-6)(-6)-2(-1) & = 38 \\ P_3 = r^3+s^3 = S_1P_2 - S_2P_1 = (-6)(38)+(-6) & = -234 \\ P_4 = r^4+s^4 = S_1P_3 - S_2P_2 = (-6)(-234)+(38) & = 1442 \\ P_5 = r^5+s^5 = S_1P_4 - S_2P_1 = (-6)(1442)+(-234) & = \boxed{-8886} \end{array} $

Sir, if it was $r^9+s^9$ or such bigger number, would we still need to individually find $r^1+s^1$ , $r^2+s^2$ and so on..? Is there a more generalised formula for the Newtons Sums?

Aman thegreat - 3 years, 3 months ago

All I know is generally we have to go through each $P_n$ progressively. There is no easier Newton Sums method. I can use an Excel spreadsheet to calculate the coefficients.

Chew-Seong Cheong - 3 years, 3 months ago

Can I use a calculator? Please?

The answer is -8886.

1 solution

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