The complex numbers ${{z}_{1}}=2$ and ${{z}_{2}}=\,\,\mathbf{i}$ are roots of the cubic equation

Question

The complex numbers ${{z}_{1}}=2$ and ${{z}_{2}}=\,\,\mathbf{i}$ are roots of the cubic equation ${{z}^{3}}+A{{z}^{2}}+Bz+C=\mathrm{ }0$, where $A$, $B$ and $C$ are real constants. State the third root and find the values of $A$, $B$ and $C$.[3] Hence, with a suitable substitution, find the roots of the equation $2w^3-w^2+2w-1=0$.[3]

Accepted Answer

(i) Third root $-i$; $A=-2,\ B=1,\ C=-2$. (ii) $w=\dfrac12,\ i,\ -i$.

2013 SRJC P1 Q4

2013 SRJC P1 Q4

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