MathDB
Complex Roots

Source: 2021 Fall AMC 12B #14

November 17, 2021
AMCAMC 12AMC 12 B

Problem Statement

Suppose that P(z),Q(z)P(z), Q(z), and R(z)R(z) are polynomials with real coefficients, having degrees 22, 33, and 66, respectively, and constant terms 11, 22, and 33, respectively. Let NN be the number of distinct complex numbers zz that satisfy the equation P(z)Q(z)=R(z)P(z) \cdot Q(z)=R(z). What is the minimum possible value of NN?
<spanclass=latexbold>(A)</span>0<spanclass=latexbold>(B)</span>1<spanclass=latexbold>(C)</span>2<spanclass=latexbold>(D)</span>3<spanclass=latexbold>(E)</span>5<span class='latex-bold'>(A)</span>\: 0\qquad<span class='latex-bold'>(B)</span> \: 1\qquad<span class='latex-bold'>(C)</span> \: 2\qquad<span class='latex-bold'>(D)</span> \: 3\qquad<span class='latex-bold'>(E)</span> \: 5