A polynomial can be though of as a mathematical expression with many factors. By definition, a polynomials is a mathematical expression consisting of only addition, subtraction, and multiplication with variables that have non-negative integer exponents. We're familiar with the ins and outs of basic polynomials like linear functions (lines) and quadratics. We can add, subtract, multiply and (in some cases) divide them (factor).
Polynomials are a large part of our mathematical vocabulary because they are used to explain and predict countless phenomena in the real world. In maths classes we tend to focus on the basics of polynomials, but the underlying ways that we can construct and deconstruct polynomials hold for the more complex models used by scientists and mathematicians to help inform policy and decisions that can have tremendous impact on the world we know.
The basic arithmetic we perform on polynomials is nothing new. Addition and subtraction of polynomials consists of combining like terms. Multiplication of polynomials is just distribution. A common initialism you've probably heard of for multiplication of polynomials is FOIL, wherein you multiply the first, outer, inner and last terms of a pair of linear factors [eg, (x+2)(x-3)] in order to find the product. While FOIL only applies to the multiplication of two linear factors, the procedure - distributing each term of the first factor into each term of the second factor - applies to the multiplication of higher degree polynomials as well. Important to note is that the sum, difference, and product of two polynomials is always just another polynomials. Also worth noting is that when multiplying polynomials of different degrees, the degree of the product polynomial will always be the sum of the degrees of the factor polynomials. For example, in (x+2)(x-3), both factors have a degree of 1, so when they are multiplied together, the product polynomial will have a degree of 1+1, or 2. This is true for all polynomial multiplication. Importantly - this tells us that the degree of a polynomial also tells us "the number of things that needed to be multiplied together" to achieve the product shown. This will be important later.
The division of polynomials, unlike addition, subtraction, and multiplication, can be a bit more involved. The information we can garner from the division of polynomials also tells us very significant information about the dividend, divisor, and quotient polynomials.
In the video above we learn conceptually about the division of polynomials and start to look at the procedure behind what is known as polynomial long division. In polynomial long division, just like with any division problem, there are four parts: the dividend (what is being divided), the divisor (what we are dividing by), the quotient (what the answer is), and the remainder. Important to note is that polynomial long division will work even when the divisor is a polynomial of degree greater than one. For example we can divide a 3rd degree polynomial (a cubic) by a 2nd degree polynomial (a quadratic), and we get a quotient that is a first degree polynomial. This makes sense, because if we consider a very simple case, like (x^3)/(x^2), we know that this is just x^1. In polynomial division, the degree of the quotient will always be equal to the degree of the dividend minus the degree of the divisor. If we had a 5th degree polynomial and divided it by a quadratic, the quotient would be a 3rd degree polynomial (5-2=3).
When we divide polynomials and are left with no remainder, we can say that the the divisor is a factor of the dividend. As shown in a previous video, factors of a polynomial correspond to zeros of a polynomial. For example, if a polynomial of high degree has a linear factor of (x-2), it means that if we plugged 2 in to the polynomial, we would get 0 out. The graph of that polynomial has an x-intercept, also called a real zero, at x=2. This is known as the Factor Theorem. Additionally, if we divide a high degree polynomial by a linear polynomial (x-a) and end up with a remainder, it not only means that that linear polynomial is not a factor of the original polynomial, but that that remainder is the result when you evaluate the original polynomial, like you would a function, for a [f(a)=remainder]. This is known as the Remainder Theorem. These can be extremely useful if we want to know whether one polynomial is a factor of another without having to tedious long division, as well as in other situations.
Simplifying the evaluation process of a polynomial function by re-writing in nested form and applying the remainder theorem to the result reveals the math behind Synthetic Division, a quick and powerful alternative to polynomial long division. Despite it's main limitation, that it can only be used for dividing polynomials by linear polynomials, this tends to be the preferred method for students. Synthetic division, combined with the factor and remainder theorem and our traditional methods for factoring certain polynomials (usually quadratics), allows us to effectively re-write high-degree polynomials in factored form. When we figure out what things (factors) needed to be multiplied together to form a complicated-looking polynomial, it reveals a lot of very useful information, primary among them being the real zeros.
Practice problems for polynomial division: PG 371 #3, 12, 27, 53 (TUSD login required to view document)
Practice problems for Factor and Remainder Theorems: PG 377 #14, 27, 38, 48 (TUSD login required to view document)
Finding the remaining factors of a higher degree polynomial isn't necessarily difficult if we've been given one or more or the factors to start with; but what if we weren't? By finding the zeros of some simple quadratics and examining the result, we find that the rational zeros of a polynomial function must come from some combination of the factors of the constant term and the coefficient of the leading term. The Rational Zeros Theorem states that if f(x) is a polynomial function with integral coefficients (not fractions), and p/q is a factor of that f(x), then p must be a factor of the constant term of f(x) and q must be a factor of the leading coefficient of f(x). By listing out all possibilities for p/q for a given polynomial, we can generate the set of possible rational zeros for that polynomial. We can systematically check if any of these possibilities are indeed zeros by using the Factor Theorem (evaluate the function for that number, a, and if the result is 0, then (x-a) is a factor), or by using synthetic division (if the remainder is 0...).
There are two limitations with this. First, the list that we generate can be quite long. Second, this only applies to rational, real zeros. It can be the case where a polynomial has neither real zeros or rational zeros (or some combination of all three).
Practice problems for Rational Zeros Theorem: PG 389 #6, 12 (TUSD login required to view document)
To address some of these limitations, we have two very important mathematical theorems: The Fundamental Theorem of Algebra, and The Conjugate Zeros Theorem. The first, as its name suggests, is very fundamental to many things in algebra. It states, simply, that a polynomial function of the nth degree has exactly "n" complex zeros. For example, a 12th degree polynomial will have 12 complex zeros. Introducing the word "complex" into our discussion invites the, "What do you mean, "complex"?" question. As we know, all numbers are complex numbers. Thus far when talking about polynomials we've only discussed real complex numbers. Non-real complex numbers, those with a non-zero imaginary component, can also be "zeros", of a polynomial function. To describe how this works we have the Conjugate Zeros Theorem, which states that if the complex number a+bi, where a and b are real numbers and b does not equal zero [in other words, a complex number with a non-zero imaginary part], is a zero of a polynomial, then a-bi must also be a zero. The reason for this is these type of solutions always come in pairs. Imagine trying to solve for x in the equation x^2=-1. The answer is ±i. You will never encounter a+bi without a-bi. They are the result of solving a quadratic with a negative discriminant (b^2 - 4ac)
Descartes' Rule of Signs, named after the most handsome man in mathematics, is helpful in informing us how many of each kind of zero we should look for once we've identified our possible zeros using the Rational Zeros Theorem.
So with all of these tools now at our disposal, we can find all of the zeros of most polynomials; even ones that are as horrible as this:
There are a few instances where we don't quite know how to do that yet, and really the reason for that is the process that we can use to do that by hand is...horrible. It incorporates something called the Intermediate Value Theorem (discussed in the previous video) and a binary search procedure called "the bisection method". Instead of that nonsense, we'll use a calculator. Here is the desmos graph associated with the video below.
Practice problems for finding zeros of a polynomial: PG 390 #13, 17, 35, 47 (TUSD login required to view document)
Now that you've learned a ton of information about polynomials, you should take some time to consider if everything you've learned makes sense. This is a topic that it is very iterative. We can start with simple examples and simple skills then add in a considerable amount of complexity as we go, but at the end, every little skill we learn has a logical justification.
When you're ready, click here to take the quiz for Week 1. You have until Sunday, April 5th, at 11:59PM.
Below is a repository of files on the topics discussed this week. Feel free to browse and use as necessary.