1 Further complex numbers: powers and roots 9
a Solve the equation z4 = −16, giving your answers in Cartesian form. b Hence express z4 + 16 as a product of two real quadratic factors.
10 a Find all the solutions of the equation z3 = −8i. b Hence solve the equation w3 + 8i(w − 1) 3 = 0. Give your answers in exact Cartesian form. 11 Consider the equation z3 + (4√2 − 4√2i) = 0. a Solve the equation, giving your answers in the form r(cos q + i sin q ). The solutions are represented on an Argand diagram by points A , B and C, labelled clockwise with A in the first quadrant. D is the midpoint of AB and the corresponding complex number is d. b Find the modulus and argument of d. c Write d3 in exact Cartesian form.
b Expand (x + 2) 3.
pl e
12 a Find, in exponential form, the three solutions of the equation z3 = −1.
c Hence or otherwise solve the equation z3 + 6z2 + 12z + 9 = 0, giving any complex solution in exact Cartesian form.
Section 4: Roots of unity
Sa m
In Section 3 you learnt a method for finding all complex roots of a number. A special case of this is solving the equation zn = 1. Its solutions are called roots of unity. WORKED EXAMPLE 1.10
Find the fifth roots of unity, giving your answers in exponential form. Let the roots be z = r eiq . Then: (r eiq) 5 = 1
Write z in exponential form and use De Moivre’s theorem.
⇒ r5e5iq = 1e0i
1 has modulus 1 and argument 0.
Comparing the moduli: r=1
Comparing the arguments: 5q = 0, 2π, 4π, 6π, 8π q = 0, 2π , 4π , 6π , 8π 5 5 5 5
Remember that there should be five solutions.
2πi
4πi
6πi
8πi
The fifth roots of unity are:1, e 5 , e 5 , e 5 , e 5
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