Difference between revisions of "Newton's Sums"
m |
Archimedes1 (talk | contribs) m (→Basic Usage) |
||
| Line 3: | Line 3: | ||
==Basic Usage== | ==Basic Usage== | ||
| − | Consider a polynomial <math>P(x)</math> of degree <math>n</math>, | + | Consider a polynomial <math>\displaystyle P(x)</math> of degree <math>n</math>, |
| − | <center><math>\displaystyle P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center> | + | <center><math>\displaystyle <math>\displaystyle P(x)</math> = 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 following sums: | Let <math>P(x)=0</math> have roots <math>x_1,x_2,\ldots,x_n</math>. Define the following sums: | ||
| Line 29: | Line 29: | ||
<math>\vdots</math> | <math>\vdots</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>\displaystyle P(x) = x^3 + 3x^2 + 4x - 8</math>. Let the roots of <math>\displaystyle P(x)</math> be <math>\displaystyle r, s</math> and <math>\displaystyle t</math>. Find <math>\displaystyle r^2 + s^2 + t^2</math> and <math>\displaystyle r^4 + s^4 + t^4</math> | ||
Newton Sums tell us that: | Newton Sums tell us that: | ||
| − | <math>S_1 + 3 = 0</math> | + | <math>\displaystyle S_1 + 3 = 0</math> |
| − | <math>S_2 + 3S_1 + 8 = 0</math> | + | <math>\displaystyle S_2 + 3S_1 + 8 = 0</math> |
| − | <math>S_3 + 3S_2 + 4S_1 - 24 = 0</math> | + | <math>\displaystyle S_3 + 3S_2 + 4S_1 - 24 = 0</math> |
| − | <math>S_4 + 3S_3 + 4S_2 - 8S_1 = 0</math> | + | <math>\displaystyle 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>\displaystyle S_1</math>, and then for the other variables, yields, |
| − | <math>S_1 = r + s + t = -3</math> | + | <math>\displaystyle S_1 = r + s + t = -3</math> |
| − | <math>S_2 = r^2 + s^2 + t^2 = 1</math> | + | <math>\displaystyle S_2 = r^2 + s^2 + t^2 = 1</math> |
| − | <math>S_3 = r^3 + s^3 + t^3 = 33</math> | + | <math>\displaystyle S_3 = r^3 + s^3 + t^3 = 33</math> |
| − | <math>S_4 = r^4 + s^4 + t^4 = -127</math> | + | <math>\displaystyle S_4 = r^4 + s^4 + t^4 = -127</math> |
| − | Which gives us our desired solutions, -127 | + | Which gives us our desired solutions, <math>\displaystyle 1</math> and <math>\displaystyle -127</math>. |
==See Also== | ==See Also== | ||
*[[Vieta's formulas]] | *[[Vieta's formulas]] | ||
Revision as of 20:54, 2 July 2007
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 
,
= a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math>Let 
have roots 
. Define the following sums:
Newton sums tell us that,
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 
.
