Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC testing, the AoPS Wiki is in read-only mode and no edits can be made.

Difference between revisions of "Newton's Sums"

(Fixed variable)
(→‎Statement)
Line 37: Line 37:
 
<math>P_2 = S_1P_1 - 2S_2</math>
 
<math>P_2 = S_1P_1 - 2S_2</math>
  
−
Etc, Where <math>S_n</math> denotes the <math>S^{th}</math> [[symmetric sum]].
+
Etc, Where <math>S_n</math> denotes the <math>n^{th}</math> [[symmetric sum]].
  
 
==Example==
 
==Example==

Revision as of 10:45, 5 April 2015

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 $P(x)$ of degree $n$,

$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

Let $P(x)=0$ have roots $x_1,x_2,\ldots,x_n$. Define the following sums:

$P_1 = x_1 + x_2 + \cdots + x_n$

$P_2 = x_1^2 + x_2^2 + \cdots + x_n^2$

$\vdots$

$P_k = x_1^k + x_2^k + \cdots + x_n^k$

$\vdots$

Newton sums tell us that,

$a_nP_1 + a_{n-1} = 0$

$a_nP_2 + a_{n-1}P_1 + 2a_{n-2}=0$

$a_nP_3 + a_{n-1}P_2 + a_{n-2}P_1 + 3a_{n-3}=0$

$\vdots$

(Define $a_j = 0$ for $j<0$.)

We also can write:

$P_1 = S_1$

$P_2 = S_1P_1 - 2S_2$

Etc, Where $S_n$ denotes the $n^{th}$ symmetric sum.

Example

For a more concrete example, consider the polynomial $P(x) = x^3 + 3x^2 + 4x - 8$. Let the roots of $P(x)$ be $r, s$ and $t$. Find $r^2 + s^2 + t^2$ and $r^4 + s^4 + t^4$.

Newton Sums tell us that:

$P_1 + 3 = 0$

$P_2 + 3P_1 + 8 = 0$

$P_3 + 3P_2 + 4P_1 - 24 = 0$

$P_4 + 3P_3 + 4P_2 - 8P_1 = 0$


Solving, first for $P_1$, and then for the other variables, yields,

$P_1 = r + s + t = -3$

$P_2 = r^2 + s^2 + t^2 = 1$

$P_3 = r^3 + s^3 + t^3 = 33$

$P_4 = r^4 + s^4 + t^4 = -127$

Which gives us our desired solutions, $\boxed{1}$ and $\boxed{-127}$.

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Newton%27s_Sums&oldid=69821"