Consider the equation $( E ) : z ^ { 3 } = 4 z ^ { 2 } - 8 z + 8$ with unknown complex number $z$. a. Prove that, for all complex numbers $z$, $$z ^ { 3 } - 4 z ^ { 2 } + 8 z - 8 = ( z - 2 ) \left( z ^ { 2 } - 2 z + 4 \right) .$$ b. Solve equation ( $E$ ). c. Write the solutions of equation ( $E$ ) in exponential form. The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D be the four points with respective affixes $$z _ { \mathrm { A } } = 1 + \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { B } } = 2 \quad z _ { \mathrm { C } } = 1 - \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { D } } = 1 .$$ 2. What is the nature of quadrilateral OABC? Justify. 3. Let M be the point with affix $z _ { \mathrm { M } } = \frac { 7 } { 4 } + \mathrm { i } \frac { \sqrt { 3 } } { 4 }$. a. Prove that points $\mathrm { A } , \mathrm { M }$ and B are collinear. b. Prove that triangle DMB is right-angled.
Consider the equation $( E ) : z ^ { 3 } = 4 z ^ { 2 } - 8 z + 8$ with unknown complex number $z$.
a. Prove that, for all complex numbers $z$,
$$z ^ { 3 } - 4 z ^ { 2 } + 8 z - 8 = ( z - 2 ) \left( z ^ { 2 } - 2 z + 4 \right) .$$
b. Solve equation ( $E$ ).
c. Write the solutions of equation ( $E$ ) in exponential form.
The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).
Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D be the four points with respective affixes
$$z _ { \mathrm { A } } = 1 + \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { B } } = 2 \quad z _ { \mathrm { C } } = 1 - \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { D } } = 1 .$$
2. What is the nature of quadrilateral OABC? Justify.
3. Let M be the point with affix $z _ { \mathrm { M } } = \frac { 7 } { 4 } + \mathrm { i } \frac { \sqrt { 3 } } { 4 }$.
a. Prove that points $\mathrm { A } , \mathrm { M }$ and B are collinear.
b. Prove that triangle DMB is right-angled.