Definition 4.2.1.
A proper rational function is one where the degree of the numerator is less than the degree of the denmoninator.
Section 4.2 Long Division and Factoring
Now I want to start the steps of turning the integral of an arbitrary rational function \(f(x) = \frac{p(x)}{q(x)}\) into a sum of integrals of the three types introduced in Section 4.1. There are three steps; two of these are covered in this section before I move on to the major new step in Section 4.3. First, I want to talk about polynomial division and polynomial factoring.
Subsection 4.2.1 Long Division
I need to start with a definition.
I want to change an arbitrary rational function into a proper rational function. To do this with fractions, say turning \(\frac{17}{5}\) into \(5 + \frac{2}{5}\text{,}\) the old technique was long division with remainder. The same is true for polynomials. To reduce a rational function to a proper ration function, I’m going to do long division with remainder. (I’ll present the long division algorithm, but the synthetic division algorithm also nicely extends to polynomials.) The goal is the same, to change an improper fraction like \(\frac{x^3}{x^2 +
1}\) into something equivalent but proper: \(x - \frac{x}{x^2
+1}\text{.}\)
In long division of numbers, each step focused on a place value (thousands, hundreas, tens, etc.). For polynomials, the steps will proceed by the degree of the polynomial (\(x^4\) then \(x^3\) then \(x^2\text{,}\) etc). The technique is best shown by example.
Example 4.2.2.
Consider this rational fucntion.
\begin{equation*}
\frac{x^4+3x^2+2x+4}{x^2+4}
\end{equation*}
This rational function is not proper. The degree of the numerator is \(4\) and the degree of the denomintor is \(2\text{.}\) I am going to do long division to make this a proper function. I divide \(x^4+3x^2+2x+4\) by \(x^2+4\text{,}\) working degree by degree and using the conventional algorithm. Looking at the first degrees, I ask how can \(x^2\) divide \(x^4\text{?}\) I need to multiply by \(x^2\text{,}\) so that is the first term of the quotient. Then I multiply \(x^2\) by \(x^2 + 4\text{,}\) subtract from \(x^4 + 3x^2 + 2x + 4\text{,}\) and continue the algorithm. I stop when the remaining term has degree less than two, since the denominator has degree two.
| \(x^2\) | \(+\) | \(0\) | \(-\) | \(1\) | |||||||||
| \(x^2\) | \(+\) | \(0\) | \(+\) | \(4\) | \(x^4\) | \(+\) | \(0\) | \(+\) | \(3x^2\) | \(+\) | \(2x\) | \(+\) | \(4\) |
| \(x^4\) | \(+\) | \(0\) | \(+\) | \(4x^2\) | |||||||||
| \(0\) | \(+\) | \(-x^2\) | \(+\) | \(2x\) | \(+\) | \(4\) | |||||||
| \(-x^2\) | \(+\) | \(0\) | \(-\) | \(4\) | |||||||||
| \(2x\) | \(+\) | \(8\) |
The quotient, \(x^2 - 1\) stands alone. The remainder becomes the new numerator. The result of the long division is this equation.
\begin{equation*}
\frac{x^4+3x^2+2x+4}{x^2+4} = x^2 - 1 +
\frac{2x+8}{x^2+4}
\end{equation*}
This is now the sum of a polynomial (which is easy to deal with) and a proper rational function.
Subsection 4.2.2 Factoring Polynomials
Long division was the first step. Now I will assume that all the rational function I am dealing with will be proper. The next step is factoring. More specifically, for the technique I am going to introduce in Section 4.3, I will need the denominators to be factored as much as possible. Factoring polynomials is somewhat familiar, particularly for quadratics, but I wanted to review the general theory before moving on. Let me start with some definitions.
Definition 4.2.4.
A polynomial factor of the form \((x-\alpha)\) is called a linear factor. Having a linear factor \((x-\alpha)\) is equivalent to saying that \(\alpha\) is a root of the polynomial.
The discriminant of a quadratic \(ax^2 + bx +
c\) is the expression \(b^2 - 4ac\text{.}\) This is the term under the square root in the quadratic formula; it determines whether or not the polynomial has real roots. A quadratic with negative discriminant has no real roots; such a quadratic is called irreducible. Since it has no real roots, it has no linear factor and cannot be further factors; hence irreducible. If an irreducible quadratic is a factor of a larger polynomials, it is very reasonably called an irreducible quadratic factor.
There is a nice theorem which tells me how polynomial over \(\RR^2\) factor. I present it here without proof. (It is a corollary of the Fundamental Theorem of Algebra; a theory which tells how polynomials factor over \(\CC\text{.}\))
Theorem 4.2.5.
Let \(p(x)\) be a polynomial with real coefficients. Then \(p(x)\) completes factors into linear and irreducible quadratic factors. Equivalently, there are no irreducible factors of degree three or higher; all such factors will always further break down into linear and quadratic terms.This simplifies the theory substantially. Assuming I can deal with factored polynomials, I only have to deal with linear and irreducible quadratic terms. This fact is central to the theory I am developing. Notice that the three types of integral I did in only had linear or quadratic denominators. I’m going to eventually reduce the integral of any rational function into one with only a linear or (irreducible) quadratic denominator. The fact that I can do so is a direct consequence of this theorem about factoring. I need one more definition.
Definition 4.2.6.
The number of times that a factor (linear or irreducible quadratic) shows up in a polynomial is called the multiplicity of the factor. For example, consider this factored polynomial.
\begin{equation*}
p(x) = (x-3)^4(x-5)^2(x^2 + 4x + 1)^7
\end{equation*}
In this polynomial, the linear factor \((x-3)\) has multiplicity \(4\text{;}\) the linear factor \((x-5)\) has multiplicity \(2\text{;}\) and the irreducible quadratic factor \((x^2 + 4x + 1)^7\) has multiplicity \(7\text{.}\)
This is the theory. In practice, the problem of actually finding the factors of a polynomial is a very difficult one. The quadratic formula nicely solves this problem for quadratics: I can determine whether a quadratic factors into two linear terms or remains irreducible based on the discriminant. For cubics and quartics, there are similar formulae. However, they get unwieldy very quickly. The cubic formula is already several lines of ordinary text and the quartic formula is several pages of ordinary text.
The situation of finding roots (and hence linear factors) of polynomials gets even worse past this point. For polynomials of degree five and higher, there is no formula for roots. Moreover, it’s not that I just don’t know the formula: in a major result in 19th mathematics (the Able-Ruffini Theorem) that such a formula is, in fact, entirely impossible. This theorem is part of a beautiful branch of mathemaatics called Galois theory, based on the work of Evariste Galois in the early 19th century.
For the purpose of this course, you need to know the terminology I defined above and the factorization theorem. However, this is not a course in factoring polynomials. I’m going to avoid all the technical challenges of actually finding (or approximating) polynomial factors by giving you factored denominators for all the integration problems involving rational functions.
