# Difference between revisions of "Newton's Sums"

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'''Newton sums''' give us a clever and efficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|identities]]. | '''Newton sums''' give us a clever and efficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|identities]]. | ||

− | == | + | ==Statement== |

Consider a polynomial <math>P(x)</math> of degree <math>n</math>, | Consider a polynomial <math>P(x)</math> of degree <math>n</math>, | ||

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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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+ | ==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> |

## Revision as of 23:21, 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.

## Statement

Consider a polynomial of degree ,

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 .