Complex numbers

Write in algebraic form

$$z=(2+3i)(1-2i)-(1+i{)}^2.$$

Solution available to those who have unlocked the subject.

Write in algebraic form

$$z=\frac{4-3i}{2+i}.$$

Solution available to those who have unlocked the subject.

Write in trigonometric form

$$z=-2+2i.$$

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Write in exponential form, with the principal argument,

$$z=-\sqrt{3}-i.$$

Solution available to those who have unlocked the subject.

Compute

$$(1-i{)}^{10}.$$

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Compute the cube roots of

$$w=-27i.$$

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Solve in $\mathbb{C}$ the equation

$$z^4=16$$

and state which polygon the solutions form in the complex plane.

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Solve in $\mathbb{C}$ the equation

$$z^2-4z+29=0.$$

Solution available to those who have unlocked the subject.

Solve in $\mathbb{C}$ the equation

$$z^4-2z^2-8=0.$$

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Solve in $\mathbb{C}$ the equation

$$z^3=i.$$

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Solve in $\mathbb{C}$ the equation

$$2z+3\bar{z}=10-i.$$

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Solve in $\mathbb{C}$ the equation

$$|z{|}^2+z-\bar{z}=9+6i.$$

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Describe the locus of the points $z\in \mathbb{C}$ such that

$$|z+1-2i|=3.$$

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Given that $z=i$ is a root of the polynomial

$$p(z)=z^3-2z^2+z-2,$$

determine all the roots and the factorization of $p$ into real factors.

Solution available to those who have unlocked the subject.