Multiplicity Examples

Will the polynomial f(x) = (x − 12)³(x − 6) change signs at x = 12?

x = 12 is a root for the polynomial, so we can check the multiplicity to see if it changes sign there. The multiplicity, 3, is odd, so that means it will pass through the x-axis. That's another way to say it will change sign.

How about we go for a challenge, though, and figure out if it changes from negative to positive, or the other way around? One way to do that would be to plug in nearby points to the equation, like x = 11 and 13, and compare their signs.

Uh, no thanks. We have a better way. Instead of actually doing the math and the multiplication, we'll only look at the signs.

For instance, when x = 11, or any number slightly less than 12, (x – 12) will be negative. So (x – 12)³ is also negative. (x – 6) will be positive, though. We have a negative times a positive, so f(somewhere close to but less than 12) is negative.

We already know that means that the polynomial will go from negative to positive, but we're going to check it anyway. We're feeling super diligent today.

Above 12, all of our terms are positive, proving it already. Being diligent isn't that hard, sometimes.

The polynomial y = x³ + x² – x − 1 has a zero at x = -1. Is it a zero of even or odd multiplicity?

To find the multiplicity at x = -1, we'll need to completely factor the polynomial. We can jump right into synthetic division:

Our remainder is the quadratic x² – 1. Easy money. Our whole polynomial factors out to (x − 1)(x + 1)(x + 1).

(x + 1) appears twice, so x = -1 is a zero of multiplicity 2. That's even, Steven. (Sorry if your name isn't Steven.)

A polynomial has an odd degree. It contains a root of multiplicity 2. Could those be the polynomial's only real roots?

We wonder what kind of degree the polynomial has. If it's not telling it, it must be odd and embarrassing. Maybe something like a degree in squirrel toenail analysis?

Anyway, let's think this through. We have a root with multiplicity of 2, so that counts as two real roots already. If we don't have any more real roots, then that means all the roots that are left are complex roots.

Complex roots come in pairs, but we have an even number of roots already. The Fundamental Theorem of Algebra says we will have an odd number of roots total. Even numbers don't add up to an odd number, though, so we must have another root with odd multiplicity.