Complex Numbers and Roots of Polynomials

This handout covers some background material about complex numbers
and roots of polynomials that will be needed for solving linear DE ’s.
Recall that a polynomial of degree n is an expression of the form

where are constants, and A root of (1) is a
number r such that

The multiplicity of a root r is the largest integer k such that (1) can be
expressed in the form , where Q(x) is another polynomial.
Note that in this case Q(x) must have degree n − k.
For example, consider the polynomial

This polynomial has roots r₁ = 1 of multiplicity k₁ = 2 and r₂ = 0 of
multiplicity k₂ = 3. Note that k₁ + k₂ = 5, that is, the multiplicities of
the roots add up to the degree of the polynomial. This is true in the more
general case when a polynomial can be written as a product of linear factors

then that is, the multiplicities of the roots add
up to the degree of the polynomial. A more informal way of expressing this
is to say that “P(x) has exactly n roots, counting multiplicities.”
Unfortunately, not every polynomial can be written as a product of linear
factors. Recall that a quadratic polynomial

has roots

The expression is called the discriminant of (3). If
then (3) has two distinct real roots. If then (3) has one real root
of multiplicity 2. In each of these two cases, the multiplicities
of the roots add up to the degree of the polynomial. If , then (3) is
indecomposable and has no real roots.

Could we somehow define numbers r₁, r₂ that could be considered distinct
roots of (3) in this third case? Consider the simplest example of an
indecomposable polynomial P₁(x) = x² + 1. It has no real roots. But let
us imagine a number i such that i² = −1, or, equivalently , Then
i² + 1 = 0, and also Thus i and −i would be
distinct roots of P₁(x), of multiplicity 1 each. Again, the multiplicities of
the roots of P₁(x) would add up to its degree. Moreover, if such i exists,
then we could write the roots of any polynomial (3) with as

Now let a, b be arbitrary real numbers. An expression of the form

z = a + bi, (6)

will be called a complex number. The number a is called the real part of
z and denoted by Re(z); the number b is called the imaginary part of z and
denoted by Im(z). Note that both Re(z) and Im(z) are real numbers. The
conjugate of a complex number z = a+bi is the expression Note
that the roots r₁, r₂ in (5) are conjugate complex numbers, with
and

Real numbers are considered a special case of complex numbers, more
precisely, they are exactly the complex numbers z with imaginary part
Im(z) = 0, that is, they are the complex numbers of the form a + 0i.
Note that a complex number is real if and only if

In order to consider complex numbers as bona fide numbers, we need to
define arithmetic operations on them. Let z₁ = a+bi and z₂ = c+di. Then

In order to see why multiplication is defined in this way, note that

With these arithmetic operations, the roots r₁, r₂ of any indecomposable
quadratic polynomial ax²+bx+c can be expressed as two conjugate complex
numbers r₁ = z₁ and (see formula (5)). Since every polynomial with
real coefficients can be factored as a product of linear and indecomposable
quadratic polynomials, we obtain the following:

Theorem 1 Suppose is a polynomial
of degree n with real coefficients. Then P(x) has exactly n complex
or real roots, counting multiplicities. Moreover, for each root z of P(x) of
multiplicity k that is not a real number, its conjugate is also a root of P(x)
with the same multiplicity k.

A few comments are in order. Since real numbers are also complex
numbers, we did not really need to write ”complex or real” in the statement
of the theorem; we did so for better readability. And since each real number
is a complex number z with the last sentence of the theorem is,
strictly speaking, also true when z is a real root of P(x). However, the
important point of the last sentence is that roots that are not real occur in
conjugate ate pairs of the same multiplicity.

If we al low the coefficients of the polynomial to be arbitrary complex
numbers, it still remains true that P(x) has exactly n complex roots, but
these are no longer required to appear in conjugate pairs. This more general
form of Theorem 1 is known as the Fundamental Theorem of Algebra.

How can we picture complex numbers geometrically? Real numbers are
usually represented as points on the real line, and they fill up this line.
There is no room for additional numbers such as i. The solution is to depict
complex numbers as vectors in a two-dimensional plane. The horizontal axis
will be labelled Re and called the real axis. The vertical axis will be labelled
Im and called the imaginary axis. A complex number z = a + bi will be
represented as a vector with Cartesian coordinates (a, b). In particular, all
real numbers are of the form (a, 0) and will be depicted on the real axis in
the usual way. The number i = 0+1i has coordinates (0, 1). In general, the
vertical axis contains all purely imaginary numbers, that is, numbers of the
form 0 + bi.

In this geometric interpretation, addition of complex numbers is simply
vector addition. In order to see how multiplication of complex numbers
works in the complex plane, we use representation of the vector (a, b) by
polar coordinates
This allows us to represent a complex number as

In this representation, r is called the absolute value or modulus of z and
often denoted by |z| and is called the angle or argument of z and often
denoted by arg(z). Now notice that according to (7) we obtain the product
of as

Thus multiplication in the complex plane, works by multiplying absolute
values and adding angles. Note that for real numbers these operations
coincide with our familiar concepts: For positive real numbers, the angles
are zero and the absolute values are the numbers themselves. Negative
real numbers have angles of π; if we multiply two of them, we obtain a
number with angle 2π, which looks the same as an angle of zero. That is,
we obtain a positive real number. Moreover, note that i has an absolute
value of one and an angle of in other words Thus
exactly as we “imagined” at the start
of our construction of complex numbers.

Finally, we will need a way to calculate e ^z for complex numbers. For
that, define exponention of purely imaginary numbers as follows:

Equation 10 is called Euler’s formula. Now we can define exponentiation
of arbitrary complex numbers as

Note that by (9) exponentiation of purely imaginary numbers satisfies
the property

which together with (11) implies for arbitrary real number x: