Question

What is the multiplicity of the real root of an equation that crosses/touches the x-axis once?

For example, `f(x)=x^3`, crosses the x-axis once. Does this mean that the solution x=0 has a multiplicity of 3?

Answer 1

A few observations...

Explanation:

Note that `f(x) = x^3` has the properties:

• `f(x)` is of degree `3`

• The only real value of `x` for which `f(x) = 0` is `x=0`

Those two properties alone are not sufficient to determine that the zero at `x=0` is of multiplicity `3`.

For example, consider:

`g(x) = x^3+x = x(x^2+1)`

Note that:

• `g(x)` is of degree `3`

• The only real value of `x` for which `g(x) = 0` is `x=0`

But the multiplicity of the zero of `g(x)` at `x=0` is `1`.

Some things we can say:

• A polynomial of degree `n > 0` has exactly `n` complex (possibly real) zeros counting multiplicity. This is a consequence of the Fundamental Theorem of Algebra.

• `f(x) = 0` only when `x=0`, yet it is of degree `3`, so has `3` zeros counting multiplicity.

• Therefore that zero at `x=0` must be of multiplicity `3`.

Why is the same not true of `g(x)`?

It is of degree `3`, so has three zeros, but two of them are non-real complex zeros, name `+-i`.

Another way of looking at this is to observe that `x=a` is a zero of `f(x)` if and only if `(x-a)` is a factor.

We find:

`f(x) = x^3 = (x-0)(x-0)(x-0)`

That is: `x=0` is a zero `3` times over.