Multiplicity - At A Glance

Michael Keaton, in his 1996 movie Multiplicity, is cloned repeatedly so he can find extra time for his family, himself, and more comedy hijinks. That is not what we are here to talk about.

What we're really going to talk about is: how many roots does y = (x – 1)² have? We often call the roots of a polynomial "zeros," because they are the spots where the x-intercept equals 0. So, we might want to say that there is one root here, at x = 1. That's ding-dong wrong, though.

We can write out the equation as y = (x – 1)(x – 1). That's two roots, even though they are both at x = 1. This is the basic idea behind the Michael Keaton of a root.

Er, wait, no. That's the multiplicity of a root. If a factor is raised by an exponent, that exponent is the multiplicity of the root. That means that x = 1 has a multiplicity of 2 in our example.

Here's a graph of y = x(x – 4)³(x + 3)². We have roots with multiplicities of 1, 2, and 3. Don't forget the multiplicity of x, even if it doesn't have an exponent in plain view. It may just want to hide, but we need an accurate head count.

A zero with an even multiplicity, like (x + 3)², doesn't go through the x-axis. It just "taps" it, and then goes back the way it came. The polynomial doesn't change signs at a zero of even multiplicity.

Zeros with an odd multiplicity, like x and (x – 4)³, pass right through the x-axis and change signs. They have places to go, people to see, and there is no time for sightseeing at the x-axis.

Notice how x just passes straight through the axis, though, while (x – 4)³ flexes and bends around it? The larger an odd multiplicity gets, the more time it spends at the gym. Honestly, once they get to around 75, we can't spend any more time around them. It's just flexing 24-7-365.

Roots with even multiplicity aren't nearly as athletic, so they get flatter and flatter bottoms as their multiplicity increases. C'mon, get up off the couch sometime.

We know that a polynomial has a total number of roots, real and complex, equal to its degree. How does multiplicity fit into this?

A root with multiplicity counts as that many zeros. For example, the factor (x − 1)¹⁰⁰ counts as 100 real roots, although it "taps" on the x-axis only once. The factor (x − 1)¹⁰¹ will pass through the x-axis only once, but will count as 101 real roots. If you ever find yourself with 101 Michael Keatons, call the fire department. That much Michael Keaton is a fire hazard.