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== | ||
+ | |||
+ | 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 | ||
+ | |||
+ | <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, | ||
+ | |||
+ | |||
+ | <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, | ||
+ | |||
+ | |||
+ | <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
,
![$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$](http://latex.artofproblemsolving.com/d/a/4/da42fa5fa45147429584e2685f8dc2e22247c497.png)
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
.