Algebra Answers

8. Express the roots of the equation z^3 − α^3 = 0 in terms of α and w, where w is a complex cube root of unity. Use your answer to find the roots of the following equations in the form a + ib.
(a) z^3 − 27 = 0
(b) z^3 + 8 = 0
(c) z^3 − i = 0.

Determine the complex number z which satisfies the equations |z + 3i| = |z + 5 −2i|
and |z − 4i| = |z + 2i| simultaneously.

4. If the roots of the cubic az
+ bz + cz + d = 0 form an arithmetic progression α − β,
2
2
α, α + β, prove that (2b
− 9ac)b + 27a
d = 0.

Show that z = i is a root of the equation z
+ z + z − 1 = 0. Find the three
other roots.

) Given that 3 + i is a root of the equation z − 3z − 8z + 30 = 0, find the
remaining roots.

What is the answer evaluate the function y=3(1/2)* for x=-3.

Find the moduli and principal arguments of w, z, wz and w/z, given that
(a) w = 10i, z = 1 + i√3, (b) w = −2√3 + 2i, z = 1 − i, (c) w = 1
1 + i√3,
1

Simplify the following expressions:
(a) (cos π/4 + i sin π/4)(cos 3π/4 + i sin 3π/4), (b) (cos π/4 + i sin π/4)2
(cos π/6 + i sin π/6)

. Find the modulus and the principal argument of each of the given complex numbers.
(a) 3 − 4i, (b) −2 + i, (c) 1
1 + i√3, (d) −74−−i3i,
(e) 5(cos π/3 + i sin π/3), (f) cos 2π/3 − sin 2π/3

. Given that w denotes either one of the non-real roots of the equation z3 = 1, show
that
(a) 1 + w + w2 = 0, and
(b) the other non-real root is w2. Show that the non-real roots of the equation
1 −u u3 can be expressed in the form Aw and Bw2, where A and B are real
numbers, find A and B.