Multiplicity of roots

Most students are familiar with the notion of multiplicity of roots for polynomials. In some situation there is a need of an analogous notion for other functions as well. Before we show two general definitions, we investigate how it works for polynomials to see where those general ideas come from.

A number c is a root of multiplicity k of a polynomial p if we can factor the term (x − c)^k out of p but we cannot factor out (x − c)^k+1. For instance, the number 1 is a root of multiplicity 2 of p(x) = x⁴ − 3x² + 2x, since

p(x) = (x − 1)²(x² + 2x²)

and we cannot factor another (x − 1) out of the second term, which is easy to see from the fact that when we put 1 into it, we do not get 0.

Obviously this is not the way to do it for general functions, since for instance 0 is a root of the function f (x) = 1 − cos(x), but we cannot factor x out of f. Thus we have to look at the procedure we use for polynomials a bit closer.

How do we show that a number c is a root of multiplicity k for a polynomial p? We divide p by (x − c)^k and call the resulting ratio g. We substitute c into g and if the claimed multiplicity is right, we get a non-zero number. If the guessed number k is lower than actual multiplicity, we get 0 after substituting c into g. If this number is higher than multiplicity, then substituting c into g leads to trouble. Indeed, if we assumed (incorrectly) that multiplicity in the above example is 3, then g would be (x² + 2x²)/(x − 1) and substituting 1 into it leads to division by zero.

If f is a function and c is a root of suspected multiplicity k, then we can try the trick with dividing by (x − c)^k and get some g as above. However, substituting c into g leads to trouble in most cases, regardless whether we guess right or not, since unlike the case with polynomials, here we cannot expect any cancelling in g. For instance, we noted above that x = 0 is a root of f (x) = 1 − cos(x), but when we divide, we get

g(x) = (1 − cos(x))/x.

This problem can be sidestepped if we use limit instead of substituting.

Definition.
Let c be a root of a function f. We say that it is a root of multiplicity k if the function

g(x) = f (x)/(x − c)^k

has a limit at c that converges to a non-zero number.

Root of multiplicity 1 is also called a simple root.

Example: Determine the multiplicity of the root 0 for f (x) = 1 − cos(x).

Could it be a simple root?

No, the root is not simple. The fact that the limit is zero shows that there is another 0 hidden in g as a root. Thus we expect that 0 is a root of multiplicity at least 2.

The convergence shows that multiplicity is at least 2 indeed, and the fact that the result is not zero shows that it is exactly 2. Thus 0 is a root of multiplicity 2 for the given function f.

This procedure is reasonable but there is a way to make it shorter. Note that we can repeat using l'Hospital's rule as many times as the numerator is zero, that is, as long as f and its derivatives are zero at c. This inspires another, somewhat more practical definition.

Definition.
Let c be a root of a function f. We say that it is a root of multiplicity k if f (c) = 0, f ′(c) = 0, ... f ^(k−1)(c) = 0, but f ^(k)(c) ≠ 0.

How does it work for the above example?

f (x) = 1 − cos(x),     f (0) = 0;
f ′(x) = sin(x),     f ′(0) = 0;
f ′′(x) = cos(x),     f ′′(0) = 1.

This confirms that 0 is indeed a root of multiplicity 2.

This example is typical, determining multiplicity of a root using derivatives is usually the best way to go. However, the first definition is more general, since it does not require differentiability at c.

Remark: For polynomials it is easy to see why this new definition follows from the first one. We will show it for multiplicity 2. Assume that c is a root of multiplicity 2 for a polynomial p, therefore p(x) = (x − c)²g(x) and g(c) is not zero. Then

p(c) = 0;
p′(x) = 2(x − c)g(x) + (x − c)²g′(x),     p′(c) = 0;
p′′(x) = 2g(x) + 4(x − c)g′(x) + (x − c)²g′′(x),     p′′(c) = 2g(c) ≠ 0.

Higher multiplicities work the same but one has to use the Leibniz formula for the derivative of product.