Difference between revisions of "Newton's Sums"
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<center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center> | <center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center> | ||
− | Let <math>P(x)=0</math> have roots <math>x_1,x_2,\ldots,x_n</math>. Define the | + | Let <math>P(x)=0</math> have roots <math>x_1,x_2,\ldots,x_n</math>. Define the sum: |
− | < | + | <cmath>P_k = x_1^k + x_2^k + \cdots + x_n^k.</cmath> |
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Newton's sums tell us that, | Newton's sums tell us that, | ||
− | < | + | <cmath>a_nP_1 + a_{n-1} = 0</cmath> |
− | + | <cmath>a_nP_2 + a_{n-1}P_1 + 2a_{n-2}=0</cmath> | |
− | < | + | <cmath>a_nP_3 + a_{n-1}P_2 + a_{n-2}P_1 + 3a_{n-3}=0</cmath> |
− | + | <cmath>\vdots</cmath> | |
− | < | ||
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− | < | ||
(Define <math>a_j = 0</math> for <math>j<0</math>.) | (Define <math>a_j = 0</math> for <math>j<0</math>.) | ||
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We also can write: | We also can write: | ||
− | < | + | <cmath>P_1 = S_1</cmath> |
− | + | <cmath>P_2 = S_1P_1 - 2S_2</cmath> | |
− | < | ||
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]]. |
Revision as of 09:22, 12 June 2021
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
[hide]Statement
Consider a polynomial of degree ,
Let have roots . Define the sum:
Newton's 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's Sums tell us that:
Solving, first for , and then for the other variables, yields,
Which gives us our desired solutions, and .