Introducing the Polynomial

I’ve done as much as I want with piecewise constant interpolations, at least for the moment. The next step that makes sense to me is to look into polynomials. They’re a powerful tool to use in interpolations, but that doesn’t stand out, because they’re powerful tools for most uses. They’re very popular mathematically, since a few polynomials can turn what was a young student’s natural interest in mathematics into a passionate lifelong loathing, with the occasional dream of being haunted by the “quadratic formula”. It’s worth taking a few paragraphs to see what polynomials are, and why they’re popular among those who get past that class.

Anyone staring at the word “polynomial” and thinking about what it might mean probably spots right away the “poly” root, suggesting, “a bunch of things”; the “nom” is a bit less obvious but there’s certainly something about “names” or “named things” swirling around there. The “ial”, well, that’s a nice math-y way of rounding out a word. So, all right, what things?

The way I’m most comfortable with to set things up is to start with a number. I don’t know or care which number it is, so, I’ll name it x. If I multiply x by x, I get the number x^2. If I multiply x by x and by x again, I get x^3. And so on; I can keep multiplying this pile by x again and again and get x^4, or x^5 or x raised to whatever number I like. This whole pile of the number x multiplied by itself is the collection of powers of x, all the way up to some highest power that’s in play for whatever the problem is. For example, if I learn that what I was calling x is actually 3, then, x^2 is 3 times 3 or 9; then x^3 is 3 times 3 times 3 or or 27; x^4 is 81, x^5 is 243, and I could carry this on until I lose interest in multiplying things by three, somewhere around the number x^n, or for this case 3^n. If you get past 2,187 you’re more indefatigable than I am.

I slipped in a bit of shorthand in the use of n there. I don’t want to tie myself down to talking about a polynomial with powers of x up to the third, or the fourth, or the tenth, or the hundredth term. So, I’ll let n be whatever number is the highest power in play here. It’s a whole number, a positive number, the sort of thing you can count up to.

The point of getting this collection of powers of x is that I have along with it a list of numbers, called the coefficients. I don’t want to say what they are right now, so I need some way to represent the numbers. I’ll need one coefficient for each power of x that I want to use, though, plus one more, and I don’t necessarily know how many that is. So we’ll fall back on using subscripts: a_1 is one coefficient, a_2 is another, a_3 is another yet, and so on, all the way up to some number a_n; we’ll also add in a_0. The n is the highest power that x gets raised to, and will be as big or as small as I like — even zero, if I really want — but it’ll never be negative and it’ll never be anything but a whole number. And all these different coefficients might be positive numbers, might be negative, might be zero, might be different from one another, might be the same as some other.

Now here’s the polynomial, in this setup at least. It’s a summation. Start out with a_0. Now add to that the product a_1 times x. Now add to that the product a_2 times x^2. And then add to that total whatever comes from a_3 times x^3. And then add what you get from a_4 times x^4. And so on, until you’ve used up all the coefficients and the powers of x.

To write it all out, the polynomial is the sum

a_0 + a_1 \cdot x + a_2\cdot x^2 + a_3 \cdot x^3 + a_4 \cdot x^4 + \cdots + a_{n - 1}\cdot x^{n - 1} + a_n \cdot x^n

That’s got some shorthand involved, as these kinds of notation always do. The \cdots part means “keep doing what we were doing the last couple steps”, an instruction which assumes that we have a pattern easy to identify already set up and that we can keep doing. In principle, this isn’t logically sound; we can set up many different patterns that all start and end the same way, so there’s not a way to pick one out. In practice, we’re not trying to pull some trick on the reader. We’re trying to communicate, so the patterns are meant to be the most obvious ones. The first term hidden by the ellipses there is a_5 \cdot x^5, and the term after that a_6 \cdot x^6, and if you don’t have strong guesses about the last term omitted there you’re probably being difficult instead.

For an example. Let’s suppose I want to talk about polynomials with powers n going up to three. Then I need four coefficients: a_0, a_1, a_2, a_3. And my polynomial will be a_0 + a_1\cdot x + a_2\cdot x^2 + a_3\cdot x^3.

Still abstract. Let’s stick with this polynomial of the same length, and … oh, let’s say a_0 should be the number 4, and a_1 the number 3, and a_2 the number 0, and a_3 the number -2. We can deal with non-whole numbers later. In that case, we have the polynomial 4 + 3 x + 0 x^2 - 2x^3.

If we decide that we want to know what value this has when x is some particular number — let’s say when x = 1 — then we’re “evaluating” the polynomial at this x = 1. Often that’s called “evaluating at the point x = 1”, by people already thinking of how we would draw a graph representing the polynomial, and thinking of what points on this graph have the x-coordinate of 1.

When we replace all the appearances of x in this polynomial with 1, we get the number, 4 + 3\cdot 1 + 0 \cdot 1^2 - 2\cdot 1^3, or 4 + 3\cdot 1 + 0 -2 \cdot 1, or 5.

Suppose we try this at the point x = -1.5 instead. Then we go through the same routine, though, replacing all appearances of x in the polynomial with -1.5. This gives a number of 4 + 3\cdot -1.5 + 0 \cdot 1.5^2 - 2\cdot -1.5^3, or 4 - 4.5 + 0 - 1.5 or -2. For any number we might pick for x, we get another number out of the polynomial.

So that’s what we want. The next question is, why do we want it?