# Difference between revisions of "Newton's Sums"

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etc., where <math>S_n</math> denotes the <math>n</math>-th [[elementary symmetric sum]]. | etc., where <math>S_n</math> denotes the <math>n</math>-th [[elementary symmetric sum]]. | ||

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+ | ==Proof== | ||

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+ | Let <math>\alpha,\beta,\gamma,...,\omega</math> be the roots of a given polynomial <math>P(x)=a_nx^n+a_{n-1}x^{n-1}+..+a_1x+a_0</math>. Then, we have that | ||

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+ | <math>P(\alpha)=P(\beta)=P(\gamma)=...=P(\omega)=0</math> | ||

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+ | |||

+ | Thus, | ||

+ | |||

+ | |||

+ | <math>\begin{cases}a_n\alpha^n+a_{n-1}\alpha^{n-1}+...+a_0=0\\a_n\beta^n+a_{n-1}\beta^{n-1}+...+a_0=0\\~~~~~~~~~~~~~~~~~~\vdots\\a_n\omega^m+a_{n-1}\omega^{n-1}+...+a_0=0\end{cases}</math> | ||

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+ | Multiplying each equation by <math>\alpha^{k-n},\beta^{k-n},...,\omega^{k-n}</math>, respectively, | ||

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+ | |||

+ | <math>\begin{cases}a_n\alpha^{n+k-n}+a_{n-1}\alpha^{n-1+k-n}+...+a_0\alpha^{k-n}=0\\a_n\beta^{n+k-n}+a_{n-1}\beta^{n-1+k-n}+...+a_0\beta^{k-n}=0\\~~~~~~~~~~~~~~~~~~\vdots\\a_n\omega^{n+k-n}+a_{n-1}\omega^{n-1+k-n}+...+a_0\omega^{k-n}=0\end{cases}</math> | ||

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+ | <math>\begin{cases}a_n\alpha^{k}+a_{n-1}\alpha^{k-1}+...+a_0\alpha^{k-n}=0\\a_n\beta^{k}+a_{n-1}\beta^{k-1}+...+a_0\beta^{k-n}=0\\~~~~~~~~~~~~~~~~~~\vdots\\a_n\omega^{k}+a_{n-1}\omega^{k-1}+...+a_0\omega^{k-n}=0\end{cases}</math> | ||

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+ | Sum, | ||

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+ | |||

+ | <math>a_n\underbrace{(\alpha^k+\beta^k+...+\omega^k)}_{S_k}+a_{n-1}\underbrace{(\alpha^{k-1}+\beta^{k-1}+...+\omega^{k-1})}_{S_{k-1}}+a_{n-2}\underbrace{(\alpha^{k-2}+\beta^{k-2}+...+\omega^{k-2})}_{S_{k-2}}+...+a_0\underbrace{(\alpha^{k-n}+\beta^{k-n}+...+\omega^{k-n})}_{S_{k-n}}=0</math> | ||

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+ | Therefore, | ||

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+ | |||

+ | <math>\boxed{a_nS_k+a_{n-1}S_{k-1}+a_{n-2}S_{k-2}+...+a_0S_{k-n}=0}</math> | ||

+ | |||

==Example== | ==Example== |

## Revision as of 11:42, 12 January 2018

**Newton sums** give us a clever and efficient way of finding the sums of roots of a polynomial raised to a power. They can also be used to derive several factoring identities.

## Contents

## Statement

Consider a polynomial of degree ,

Let have roots . Define the following sums:

Newton sums tell us that,

(Define for .)

We also can write:

etc., where denotes the -th elementary symmetric sum.

## Proof

Let be the roots of a given polynomial . Then, we have that

Thus,

Multiplying each equation by , respectively,

Sum,

Therefore,

## Example

For a more concrete example, consider the polynomial . Let the roots of be and . Find and .

Newton Sums tell us that:

Solving, first for , and then for the other variables, yields,

Which gives us our desired solutions, and .