It is given that $-2+2i$ is a root of the equation ${{z}^{3}}+a{{z}^{2}}+bz-16\sqrt{2}=0$, where $a$

Question

It is given that $-2+2i$ is a root of the equation ${{z}^{3}}+a{{z}^{2}}+bz-16\sqrt{2}=0$, where $a$ and $b$ are real numbers. Find the values of $a$ and $b$ and the other two roots. Leave your answers in the exact form. [5] In an Argand diagram with origin $O$, the three roots are represented by points $A$, $B$ and $C$ where $A$ represents $-2+2i$ and $C$ represents the real root. Label these points on an Argand diagram, indicating clearly the modulus and argument of each root. State also a geometrical relationship between $A$ and $B$. [3] Hence, prove that $	an \frac{3\pi }{8}=1+\sqrt{2}$. [3]

Accepted Answer

a = 4 - 2√2, b = 8 - 8√2, other roots: -2 - 2i and 2√2; tan(3π/8) = 1 + √2

2025 TMJC P1 Q9

2025 TMJC P1 Q9

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