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cis(120) * r is also a root

Source: 2020 AMC 12B #17

February 6, 2020
algebrapolynomial2020 AMC 12B2020 AMCAMC 12AMC 12 BAMC

Problem Statement

How many polynomials of the form x5+ax4+bx3+cx2+dx+2020x^5 + ax^4 + bx^3 + cx^2 + dx + 2020, where aa, bb, cc, and dd are real numbers, have the property that whenever rr is a root, so is 1+i32r\frac{-1+i\sqrt{3}}{2} \cdot r? (Note that i=1i=\sqrt{-1})
<spanclass=latexbold>(A)</span>0<spanclass=latexbold>(B)</span>1<spanclass=latexbold>(C)</span>2<spanclass=latexbold>(D)</span>3<spanclass=latexbold>(E)</span>4<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> 4