<p>There are real numbers <img alt="$a, b, c,$" height="18" src="https://latex.artofproblemsolving.com/2/d/0/2d02bb2275100261109b043261e702124460a83e.png" width="48" /> and <img alt="$d$" height="14" src="https://latex.artofproblemsolving.com/9/6/a/96ab646de7704969b91c76a214126b45f2b07b25.png" width="11" /> such that <img alt="$-20$" height="12" src="https://latex.artofproblemsolving.com/1/f/1/1f1588c7ab561762e3d7b81887b584140570fd01.png" width="31" /> is a root of <img alt="$x^3 + ax + b$" height="16" src="https://latex.artofproblemsolving.com/b/e/a/bea08dc21e87ebb9f32539cf2fcc9059cb22dfdc.png" width="90" /> and <img alt="$-21$" height="12" src="https://latex.artofproblemsolving.com/4/8/1/481aa8fd3205ddca64b0c5f46dab544c8a74aa36.png" width="31" /> is a root of <img alt="$x^3 + cx^2 + d.$" height="16" src="https://latex.artofproblemsolving.com/4/9/0/49077b7393dbd2ab41fa9b056821c09770282ef9.png" width="101" /> These two polynomials share a complex root <img alt="$m + \sqrt{n} \cdot i,$" height="18" src="https://latex.artofproblemsolving.com/4/9/0/490d46a3531441c7b3d4caeef49dca10c6b78e8a.png" width="88" /> where <img alt="$m$" height="10" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="18" /> and <img alt="$n$" height="10" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="13" /> are positive integers and <img alt="$i = \sqrt{-1}.$" height="18" src="https://latex.artofproblemsolving.com/4/a/2/4a2e92550cefae6e71b66ea8a86ac40ebcd6b9af.png" width="72" /> Find <img alt="$m+n.$" height="14" src="https://latex.artofproblemsolving.com/5/d/b/5db26f75f6218a382840f0ed34f5d36bcff61d33.png" width="56" /></p>

Question No: 439

There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i,$ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$

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Tags: Algebra

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