Complex Analysis Answers

Find the real root of the equation z3 + z + 10 = 0 given that one root is 1 − 2i.
(b) Given that 3 + i is a root of the equation z3 − 3z2 − 8z + 30 = 0, find the remaining roots.
(c) Given that 1 + i is a root of the equation z3 − 2z + k = 0, find the other two roots and the value of the real constant k.
(d) Given that 2−3i is a root of the equation z3 +pz2 +qz +13 = 0, find the other two roots and the values of the real constants p and q.
(e) Show that z = i is a root of the equation z4 + z3 + z − 1 = 0. Find the three other roots.
(f) Show that z = −1+i is a root of the equation z4 −2z3 −z2 +2z +10 = 0. Find the remaining roots.
(g) Solve the equation 6z4 − 47z3 + 148z2 − 167z + 52 = 0

Given that z = 1 + i
√
2, express in the form a + ib each of the complex numbers
p = z + 1/z, q = z − 1/z. In an Argand diagram, P and Q are the points which
represent p and q respectively, O is the orgin, M is the midpoint of P Q and G is the
point on OM such that OG =
2
3
OM. Prove that angle P GQ is a right angle.

Given that w denotes either one of the non-real roots of the equation z
3 = 1, show
that
(a) 1 + w + w
2 = 0, and
(b) the other non-real root is w
2
. Show that the non-real roots of the equation

1 − u
u
3
can be expressed in the form Aw and Bw2
, where A and B are real
numbers, find A and B

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 u
3 numbers, find A and B.
11. Given that 1, w, w2 are the cube roots of unity, find the equation whose roots are 1/3, 1/(2 + w), 1/(2 + w2).
can be expressed in the form Aw and Bw2, where A and B are real

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 u
3 numbers, find A and B.
11. Given that 1, w, w2 are the cube roots of unity, find the equation whose roots are 1/3, 1/(2 + w), 1/(2 + w2).
can be expressed in the form Aw and Bw2, where A and B are real

press the roots of the equation z3 − α3 = 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.

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

Express the roots of the equation z3 −α3 = 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.

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 az3 + bz2 + cz +d = 0 form an arithmetic progression α−β, α, α + β, prove that (2b2 − 9ac)b + 27a2d = 0.