MathDB
Polynomials & Sum

Source: 2012 AMC 12 B #23

February 23, 2012
algebrapolynomialinequalitiestriangle inequalityAMC

Problem Statement

Consider all polynomials of a complex variable, P(z)=4z4+az3+bz2+cz+dP(z)=4z^4+az^3+bz^2+cz+d, where a,b,ca, b, c and dd are integers, 0dcba40 \le d \le c \le b \le a \le 4, and the polynomial has a zero z0z_0 with z0=1|z_0|=1. What is the sum of all values P(1)P(1) over all the polynomials with these properties?
<spanclass=latexbold>(A)</span> 84<spanclass=latexbold>(B)</span> 92<spanclass=latexbold>(C)</span> 100<spanclass=latexbold>(D)</span> 108<spanclass=latexbold>(E)</span> 120 <span class='latex-bold'>(A)</span>\ 84\qquad<span class='latex-bold'>(B)</span>\ 92\qquad<span class='latex-bold'>(C)</span>\ 100\qquad<span class='latex-bold'>(D)</span>\ 108 \qquad<span class='latex-bold'>(E)</span>\ 120