Difference between revisions of "Newton's Sums"
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<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> | ||
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<math>S_4 + 3S_3 + 4S_2 - 8S_1 = 0</math> | <math>S_4 + 3S_3 + 4S_2 - 8S_1 = 0</math> | ||
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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 22:18, 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 ,
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 .