Homework 1 Due Monday, Aug. 30 1. Find the cube roots of 8i. 2. If ...

1. Find the cube roots of 8i.
Homework 1
Due Monday, Aug. 30
2. If you consider the polynomial ax2 + bx + c, the quadratic formula tells us
that the roots of this polynomial are
x = − b
2a ±
√
b2 − 4ac
.
2a
Note that if a, b, and c are real, and if the polynomial has complex roots
(i.e. b 2 − 4ac < 0), then the roots are conjugates of each other. Prove
that if z is a root of the polynomial anx n + an−1x n−1 + . . . a1x + a0 with
an, . . . , a0 ∈ 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 − 4z2
+ 3
≤ 1.
(Note that the denominator factors.)
5. Prove that if z1 and z2 are complex numbers, then |z1| = |z2| if and only
if there are complex numbers c1 and c2 such that z1 = c1c2 and z2 = c1c2.
6. The complex numbers C form a vector space of dimension 2 over the real
numbers R. One basis for this vector space is given by β = {1, i}.
(a) Let T : C → C by T (z) = iz. Prove that T is linear, and compute
the matrix for T with respect to the basis β.
(b) If A is the matrix that you found in part (a), why is it the case that
A 2 = −I? (Here I denotes the 2 × 2 identity matrix.) If A is any
n × n matrix with n odd and with real entries, why can A 2 never
equal −I? (Here I denotes the n × n identity matrix.)
7. A general cubic equation in X has the form
X 3 + AX 2 + BX + C
and its roots may be viewed as the intersection of the graph of this polynomial
with the X-axis.

(a) Show that the inflection point of this graph occurs at X = − A
3 .
(b) Deduce (geometrically) that the substitution X = (x− A
3 ) will reduce
the above equation to the form x3 + bx + c.
(c) Verify part (b) by calculation.
8. In order to find a solution to the cubic equation x 3 = 3px + 2q, do the
following:
(a) Make the inspired substitution x = s + t, and deduce that x solves
the cubic if st = p and s 3 + t 3 = 2q.
(b) Eliminate t between these two equations, thereby obtaining a quadratic
equation in s 3 .
(c) Solve this quadratic to obtain the two possible values of s 3 . By
symmetry, what are the possible values of t 3 ?
(d) Given that we know that s3 + t3 = 2q, deduce that
x = 3
q + q2 − p3 + 3
q − q2 − p3 .
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