MathDB
Sums Powers of Roots

Source: AMC 12A 2019 #17

February 8, 2019
AMCAMC 12AMC 12 Aalgebrapolynomial2019 AMC 12A2019 AMC

Problem Statement

Let sks_k denote the sum of the <spanclass=latexitalic>k</span><span class='latex-italic'>k</span>th powers of the roots of the polynomial x35x2+8x13x^3-5x^2+8x-13. In particular, s0=3s_0=3, s1=5s_1=5, and s2=9s_2=9. Let aa, bb, and cc be real numbers such that sk+1=ask+bsk1+csk2s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2} for k=2k = 2, 33, ........ What is a+b+ca+b+c?
<spanclass=latexbold>(A)</span>  6<spanclass=latexbold>(B)</span>  0<spanclass=latexbold>(C)</span>  6<spanclass=latexbold>(D)</span>  10<spanclass=latexbold>(E)</span>  26<span class='latex-bold'>(A)</span> \; -6 \qquad <span class='latex-bold'>(B)</span> \; 0 \qquad <span class='latex-bold'>(C)</span> \; 6 \qquad <span class='latex-bold'>(D)</span> \; 10 \qquad <span class='latex-bold'>(E)</span> \; 26