Note: Newton's Sums and Vieta's Formula work hand in hand, for the majority of problems. In other words, if a solution to a question has Newton's Sums, it will probably also have Vieta's Formulas in use.
Formula
For a polynomial P(x) of degree n, where
P(x)=anxn+an−1xn−1+…+a1x+a0
Let P(x)=0 have roots x1,x2,…,xn.
Then the sum would be
Pk=x1k+x2k+…+xnk.
Newton's sums basically says,
anPk+an−1Pk−1+…+an−k+1P1+k⋅an−k=0
(For ai=0 for i<0.)
Here are some practice problems.
Practice Problem
(2019 AMC 12A Q17)
Let sk denote the sum of the kth powers of the roots of the polynomial x3−5x2+8x−13. In particular, s0=3,s1=5, and s2=9.
Let a,b, and c be real numbers such that sk+1=ask+bsk−1+csk−2 for k=2,3,… What is a+b+c?
Let's try a harder problem.
Practice Problem
(2008 AIME 2 Q7)
Let r,s, and t be the three roots of the equation
8x3+1001x+2008=0.
Find (r+s)3+(s+t)3+(t+r)3.
Final Notes and Tips
- Not super common like Vieta's Formula but would still recommend memorizing or keep in the back of your head when you see a polynomial roots related question.