Algebraic Manipulation of Polynomials and Rational Functions

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. This simple definition hides a great deal of power and structure.

To get a formal introduction to polynomials and their associated terminology, please watch the following video from Professor Leonard. It provides a clear, conceptual overview.

To summarize the key points from the video:

• Standard Form: A polynomial is written in standard form when its terms are ordered by descending exponents. For example, $x^4+5x^2-3x+7$.
• Degree: The degree of a polynomial is the highest exponent of its variable. The degree tells us a lot about the shape and complexity of the polynomial's graph.
• Leading Term: In standard form, the first term is the leading term. Its behavior dominates the polynomial's behavior for very large values of $x$.
• Constraints: The exponents must be non-negative integers. This means expressions with fractional exponents (like $\sqrt{x}=x^{1/2}$) or negative exponents (like $1/x=x^{-1}$) are not polynomials.

Operations on Polynomials

Now that we have a solid definition, let's explore how to manipulate these expressions. Like numbers, we can add, subtract, multiply, and divide them.

The following video provides a practical guide with worked examples for these operations.

Addition and subtraction are straightforward applications of combining like terms. Multiplication, while more involved, is a systematic process of distribution. Division, however, is where things get more interesting and lead to deeper theoretical insights.

The Theory Behind Division: The Factor and Remainder Theorems

The videos showed you the "how" of polynomial division. As you're aiming for a research-level understanding, the "why" is equally critical. The relationship between division, factors, and the zeros of a polynomial is formalized by two important theorems.

The following document provides a rigorous look at these concepts, including their proofs, which you should find insightful.

Let's distill the essence of these theorems:

• The Remainder Theorem: If you divide a polynomial $p(x)$ by $(x-c)$, the remainder is simply $p(c)$. This gives you a quick way to evaluate a polynomial at a certain point. For example, the remainder of $(x^2+2x+5)÷(x-1)$ is $p(1)=1^2+2(1)+5=8$.

• The Factor Theorem: This is a direct consequence of the Remainder Theorem. It states that $(x-c)$ is a factor of $p(x)$ if and only if $p(c)=0$. This creates a fundamental link: finding the zeros (or roots) of a polynomial is the same as finding its linear factors.

For instance, if we use synthetic division to divide $p(x)=x^3-2x^2-5x+6$ by $(x-1)$ and find the remainder is 0, the Factor Theorem tells us that $x=1$ is a zero of $p(x)$ and $(x-1)$ is a factor.

Rational Functions and Partial Fraction Decomposition

We've seen that not every algebraic expression is a polynomial. A rational function is a function formed by the ratio of two polynomials, $\frac{P(x)}{Q(x)}$.

Sometimes, a complex rational function is easier to work with if we break it down into a sum of simpler fractions. This process is called partial fraction decomposition. This technique is indispensable in integral calculus, and its principles of decomposition are echoed in more advanced fields relevant to machine learning, such as signal processing.

First, a crucial distinction:

• If the degree of the numerator is less than the degree of the denominator, the fraction is proper.
• If the degree of the numerator is greater than or equal to the degree of the denominator, the fraction is improper. In this case, you must first perform polynomial division to get a polynomial plus a proper fraction.

The following resource explains the method for decomposing proper rational functions.

The key is to know what form the decomposition will take based on the factors of the denominator. The following table provides a convenient summary of the rules.

Let's walk through an example for the first case: distinct linear factors.
Suppose we want to decompose $\frac{x+7}{(x-1)(x+3)}$.
The denominator has distinct linear factors $(x-1)$ and $(x+3)$. According to the rules (row 1 in the table), the decomposition will have the form:

To find $A$ and $B$, we multiply both sides by the common denominator $(x-1)(x+3)$:

This equation must hold for all values of $x$. We can find $A$ and $B$ by substituting convenient values for $x$:

• To find $A$, let $x=1$. This makes the $B$ term zero: $$1+7=A(1+3)+B(1-1)⟹8=4A⟹A=2$$
• To find $B$, let $x=-3$. This makes the $A$ term zero: $$-3+7=A(-3+3)+B(-3-1)⟹4=-4B⟹B=-1$$

So, the decomposition is:

Conclusion

In this foundational lesson, we have covered a significant amount of ground. You've learned to define, identify, and manipulate polynomials, which are central to describing functions and models in science and engineering. We went beyond simple mechanics to understand the formal theorems that govern polynomial division and factoring, providing the rigorous understanding necessary for advanced study. Finally, we introduced rational functions and the powerful technique of partial fraction decomposition.

Key Takeaways:

• Polynomials are expressions with non-negative integer exponents, and their properties (like degree) determine their behavior.
• Polynomial arithmetic (addition, subtraction, multiplication, division) follows a clear set of rules.
• The Factor Theorem provides a crucial bridge between the algebraic factors of a polynomial and its geometric zeros.
• Rational functions can be broken into simpler partial fractions, a technique that simplifies complex expressions.

In our next lesson, we will shift gears slightly to another cornerstone of mathematics for machine learning: solving systems of linear equations. This will be our first formal step into the world of linear algebra, where we move from dealing with single equations to managing sets of them simultaneously—a skill that is absolutely essential for nearly every area of machine learning.