Difference between revisions of "Newton's Sums"
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==Example== | ==Example== | ||
− | 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>. |
Newton Sums tell us that: | Newton Sums tell us that: |
Revision as of 00:13, 19 February 2011
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.
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
.)
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
.