A Spreadsheet for Finding Roots of Polynomials

The best way to understand a mathematical object is to play around with it. You should experiment with its definition, push its boundaries, and observe what happens. [1]

The first complex piece of math most people learn about is the polynomial, and it's no exception to this rule. Let's see what we can figure out about it.

The Spreadsheet

Here's a spreadsheet for experimenting with polynomials.

A spreadsheet with a graph of x^3 - 5x^2 + 6x, including its roots

(You'll need DataNitro, numpy and matplotlib for it to work; please import the scripts "get_roots.py" and "functions.py" before pressing the "get roots" button.)

The sheet lets you set the degree, or highest power of x, in the polynomial, as well as the coefficients of the different powers of x. It'll graph the polynomial, and find its roots for you.

Let's look at the graphs first.

Quadratic Equations

The simplest interesting polynomial has degree two: a quadratic polynomial (like f(x) = x2 - 5x + 6). If we set a quadratic polynomial equal to zero, we get a quadratic equation. [2]

Quadratic functions are counterintuitive, at least if you're used to linear functions (like f(x) = 3x). Even the simplest one, f(x) = x2, is strange.

A linear function increases at a steady rate: whether you go from x = 1 to x = 2 or x = 101 to x = 102, f(x) = 3x will increase by 3. But x2 increases by 3 as x goes from 1 to 2, by 23 as you from 11 to 12, and by 203 as you go from 101 to 102. [3]

For another thing, x2 is always positive - something that's definitely not true for linear functions.

If you start solving quadratic equations, you'll notice something else that's peculiar - sometimes, like with x2 = 0, they have one solution, just like a linear equation.

A graph of x^2

But sometimes, they have two:

A graph of x^2 - 4

and sometimes they have none:

A graph of x^2 + 4

What's going on? Let's look at polynomials in general, and see if we can figure it out. [4]

Higher Degree Polynomials

A general polynomial is a sum of powers of x (f(x) = x3 + 5x2 + 6x + 1).

A graph of x^3 + 5x^2 + 6x + 1

Roots: The roots of a polynomial f(x) are the values of x where f is equal to zero. Why do we care about zero in particular? Well, we don't, but we do care about equality: it seems worthwhile to be able to figure out when f(x) = g(x). One of the nice things about polynomials is that you can add or subtract two polynomials to get a third one. We'll know when f is equal to g if we figure out when f(x) - g(x) is equal to 0.[5]

Here are a few things you might notice as you play with the model:

1: An nth degree polynomial has exactly n roots. [6] This doesn't agree with what we saw above, with x2 = 0 and with x2 + 4 = 0. What's going on?

2: As we saw, some quadratic equations have no roots on the graph. When we run x2 + 4 through the spreadsheet, we its roots are 2i and -2i. What does that mean?

i is called an imaginary number, and it's defined as the square root of negative one. That is, i2 = -1. [7] Our spreadsheet can't graph it because it doesn't fit on a regular number line, but if we accept it as a concept, we can see that it does solve our equation:

(2i)2 + 4 = 4(i)2 + 4 = -4 + 4 = 0.

3: In addition to imaginary roots, some polynomials, like x2 - 4x + 4, have a single root listed more than once. This is called the multiplicitly of the root, and it's not obvious what it means. If 2 is a root of the polynomial f, how do we know whether it's a root one time or twenty times?

Roots are closely related to factoring a polynomial - for example, the roots of x2 - 4x + 4 are 2 and 2, and you can write x2 - 4x + 4 as (x-2)(x-2). We say that a root r has a multiplicity greater than one if
(x-r) is a factor of the polynomial more than once.

4: If you change a polynomial's coefficients slightly, its roots will only change slight as well. For example, x3 - 5x2 + 6x has roots 0, 2, and 3, and x3 - 5x2 + 6x + 0.01 has roots -0.002, 2.005, and 2.997.

This is a result of continuity, a property of functions. The original idea behind continuity is that you can draw some functions without lifting your pen from the paper - polynomials, for example - while other functions have gaps. [8] Continuous functions are "well-behaved" in all sorts of ways, and this is one of them: if you give a continuous function a small perturbation, its roots won't change dramatically.

5: You might also notice that the roots calculated by the spreadsheet aren't exact: they're rounded to 3 decimal points. This is usually good enough - and we can get much more accurate numbers if we need to - but it'd be best to have an exact answer. We can use the quadratic formula for second degree equations to tell us that of x2 - x - 1 are exactly (1 + √5 )/2 and (1 - √5 )/2, not 1.618 and -0.618.

This leads to a natural question: can we find the exact values of the roots of any polynomial? If we have a quadratic formula, is there a cubic formula? A quartic (4th degree) formula? A quintic (5th degree) one? [9] [10]


[1] This is how real math is done: people find new theorems and decide on new definitions by playing around. Unfortunately, math isn't taught this way; I distinctly remember memorizing the quadratic formula without understanding how it worked.

[2] Constants (like f(x) = 3) and linear functions (like f(x) = 2x + 3) are polynomials too, but they're boring. Linear equations are trivial, unless you start doing interesting things like adding variables or restricting yourself to solving them with integers.

[3] I use both f and f(x) to refer to polynomials (or functions in general). f is the function itself, while f(x) represents the value of the function at one point - for example, f(2) is the value of f at 2. f(x) is one way of representing the abstract function, by describing the value it takes for any input.

You can describe functions in other ways, too; as the sum of two other functions, for example.

[4] When you're working on a problem, it can be helpful to look at a more general version. The general version is usually - though not always - harder to solve, but it can bring some patterns to light that the specific case obfuscates.

[5] This demonstrates two general concepts in mathematics: closure and reduction. Closure is a formal property of some mathematical objects under some operations: an object is closed with respect to an operation if using that operation on the object will give you another object of the same type. For example, integers are closed under addition: if you add two integers you get another integer (1 + 1 = 2). Integers are also closed under subtraction and multiplication, but not under division (2/3 isn't an integer).

Polynomials are closed under addition and subtraction, which is why when we set two polynomials equal to each other [f(x) = g(x)], we can reduce that equation to setting another polynomial equal to 0 [f(x) - g(x) = 0]. Like integers, polynomials are also closed under multiplication, but not under division; it's a good exercise to convince yourself of this. (Once you do, here's a follow-up question: how do the roots of a product of polynomials relate to the roots of its factors?)

Reduction, on the other hand, is a general problem solving technique: reducing one set of problems (when are two polynomials equal?) to a simpler set of problems (when is a polynomial equal to 0?) is a way to get more mileage from our work.

[6] This is an elegant result, known as the "Fundamental Theorem of Algebra". There are lots of different ways to prove it - if you'd like to try, I'd recommend starting with quadratic polynomials and using induction to work your way up.

[7] "Imaginary" is a misnomer. Imaginary numbers are hard to picture, but that's because we're not used to them. They're just as concrete as negative numbers, which are also "imaginary" - that is, they don't obviously correspond to things in the real world.

In fact, mathematicians debated for centuries whether negative numbers really existed - after all, you can't have a negative apple. As it turns out, though, a negative apple is a useful concept in describing what happens when you go from 4 apples to 3 apples, regardless of whether negative apples "really" exist. Imaginary numbers have similarly concrete, if slightly more esoteric, uses. Calculating properties of electrical circuits, for instance.

[8] The modern definition of continuity is much more refined.

[9] This is a hard question.

[10] Thanks for reading to the end of the post! If you have any questions, to email me: ben@datanitro.com.