Challenging +1.8 This is a Further Maths question requiring identification of cube roots of unity (1, ω, ω²), calculation of their midpoints, and construction of a degree 6 polynomial. While it involves multiple steps (finding midpoints, forming factors, expanding), the approach is methodical rather than requiring novel insight. The algebraic manipulation is substantial but follows standard techniques for Further Maths students.
9 The cube roots of unity are represented on the Argand diagram below by the points \(A , B\) and \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-8_760_800_303_244}
The points \(L , M\) and \(N\) are the midpoints of the line segments \(A B , B C\) and \(C A\) respectively.
Determine a degree 6 polynomial equation with integer coefficients whose roots are the complex numbers represented by the points \(A , B , C , L , M\) and \(N\).
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9 The cube roots of unity are represented on the Argand diagram below by the points $A , B$ and $C$.\\
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-8_760_800_303_244}
The points $L , M$ and $N$ are the midpoints of the line segments $A B , B C$ and $C A$ respectively.
Determine a degree 6 polynomial equation with integer coefficients whose roots are the complex numbers represented by the points $A , B , C , L , M$ and $N$.
\section*{END OF QUESTION PAPER}
\hfill \mbox{\textit{OCR Further Pure Core 1 2022 Q9 [5]}}