Difference between revisions of "Newton's Sums"
Etmetalakret (talk | contribs) |
Mathandski (talk | contribs) m (Not enough rows to figure out what's going on) |
||
(4 intermediate revisions by 3 users not shown) | |||
Line 24: | Line 24: | ||
<cmath>P_1 = S_1</cmath> | <cmath>P_1 = S_1</cmath> | ||
<cmath>P_2 = S_1P_1 - 2S_2</cmath> | <cmath>P_2 = S_1P_1 - 2S_2</cmath> | ||
+ | <cmath>P_3 = S_1P_2 - S_2P_1 + 3S_3</cmath> | ||
+ | <cmath>P_4 = S_1P_3 - S_2P_2 + S_3P_1 - 4S_4</cmath> | ||
+ | <cmath>P_5 = S_1P_4 - S_2P_3 + S_3P_2 - S_4P_1 + 5S_5</cmath> | ||
+ | <cmath>\vdots</cmath> | ||
− | + | where <math>S_n</math> denotes the <math>n</math>-th [[elementary symmetric sum]]. | |
==Proof== | ==Proof== | ||
Line 91: | Line 95: | ||
==Practice== | ==Practice== | ||
[https://artofproblemsolving.com/wiki/index.php/2019_AMC_12A_Problems/Problem_17 2019 AMC 12A Problem 17] | [https://artofproblemsolving.com/wiki/index.php/2019_AMC_12A_Problems/Problem_17 2019 AMC 12A Problem 17] | ||
+ | |||
+ | [https://artofproblemsolving.com/wiki/index.php/2003_AIME_II_Problems/Problem_9 2003 AIME II Problem 9] | ||
==See Also== | ==See Also== | ||
Line 98: | Line 104: | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
− | [[Category: | + | [[Category:Polynomials]] |
[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 18:07, 21 July 2022
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.
Contents
Statement
Consider a polynomial of degree ,
Let have roots . Define the sum:
Newton's sums tell us that,
(Define for .)
We also can write:
where denotes the -th elementary symmetric sum.
Proof
Let be the roots of a given polynomial . Then, we have that
Thus,
Multiplying each equation by , respectively,
Sum,
Therefore,
Note: This technically only proves the statements for the cases where . For the cases where , an argument based on analyzing individual monomials in the expansion can be used (see http://web.stanford.edu/~marykw/classes/CS250_W19/Netwons_Identities.pdf, for example.)
Example
For a more concrete example, consider the polynomial . Let the roots of be and . Find and .
Newton's Sums tell us that:
Solving, first for , and then for the other variables, yields,
Which gives us our desired solutions, and .