Homework 1
Due Monday, Aug. 30
1. Find the cube roots of 8
i
.
2. If you consider the polynomial
ax
2
+
bx
+
c
, the quadratic formula tells us
that the roots of this polynomial are
x
=

b
2
a
±
√
b
2

4
ac
2
a
.
Note that if
a
,
b
, and
c
are real, and if the polynomial has complex roots
(i.e.
b
2

4
ac <
0), then the roots are conjugates of each other.
Prove
that if
z
is a root of the polynomial
a
n
x
n
+
a
n

1
x
n

1
+
. . . a
1
x
+
a
0
with
a
n
, . . . , a
0
∈
R
, then
z
is also a root. (In other words, the complex roots
of a polynomial with real coefficients occur in conjugate pairs.)
3. Let
n
be a positive integer greater than or equal to 2.
(a) What is the sum of the
n
th
roots of 1?
(Prove that your claim is
true.)
(b) What is the product of the
n
th
roots of 1? (Again, prove your claim.)
4. Suppose that
z
is a complex number lying on the circle of radius 2 centered
at the origin (i.e.,

z

= 2). Prove that
z
+ 1
z
4

4
z
2
+ 3
≤
1
.
(Note that the denominator factors.)
5. Prove that if
z
1
and
z
2
are complex numbers, then

z
1

=

z
2

if and only
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 Fall '09
 Forde
 Quadratic Formula, Quadratic equation, Complex number

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