What does "$f(x)$ has a root of order $k$" mean?

Question

I'm confused by the wording of this, the question states:

Assume $f(x)$ has a root of order $k=3$ at $x(n)$.

What does the "root of order $k=3$" part mean?

Thanks

Answer 1

You should say it has a root of order $3$ at some point, say $n$, not $x$ which is the variable you use for the function to avoid confusion. Very roughly speaking, it means $f(x)$ looks like $(x-n)^3g(x)$ where $g(x)$ is finite and nonzero at $n$. You have $f(n)=0, f'(n)=0, f''(n)=0, f'''(0) \neq 0$. In some cases we mean it has a root of at least $3$ and you might have $f'''(n)=0$. You need to figure that out from context.