Difference between revisions of "Newton's Sums"
(categories?) |
m (→Basic Usage) |
||
Line 29: | Line 29: | ||
<math>\vdots</math> | <math>\vdots</math> | ||
+ | (Define <math>a_j = 0</math> for <math>j<0</math>.) | ||
For a more concrete example, consider the polynomial <math>P(x) = x^3 + 3x^2 + 4x - 8</math>. Let the roots of <math>P(x)</math> be <math>r, s</math> and <math>t</math>. Find <math>r^2 + s^2 + t^2</math> and <math>r^4 + s^4 + t^4</math> | For a more concrete example, consider the polynomial <math>P(x) = x^3 + 3x^2 + 4x - 8</math>. Let the roots of <math>P(x)</math> be <math>r, s</math> and <math>t</math>. Find <math>r^2 + s^2 + t^2</math> and <math>r^4 + s^4 + t^4</math> | ||
Line 41: | Line 42: | ||
<math>S_4 + 3S_3 + 4S_2 - 8S_1 = 0</math> | <math>S_4 + 3S_3 + 4S_2 - 8S_1 = 0</math> | ||
+ | |||
Solving, first for <math>S_1</math>, and then for the other variables, yields, | Solving, first for <math>S_1</math>, and then for the other variables, yields, |
Revision as of 23:18, 25 April 2010
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.
Basic Usage
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
.)
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
.