Roots of product of two functions

Question

I wonder if the answer to this question is true:

Having two functions $f(x)$, $g(x)$ where $f(x)$ has $N$ real roots, and $g(x)$ is positive for all $x$ (no real roots), does the product of $f(x)g(x)$ also have exactly $N$ roots?

For example. Let $f(x)$ is a polynomial and $g(x)=e^x$, then clearly roots of $$f(x)e^x=c(x-a_1)(x-a_2)\dots(x-a_N)e^x=0$$ are again only $a_1,a_2,\dots,a_N$.

Does this hold for general functions $f(x)$, $g(x)$ ($f(x)$ not necessarily a polynomial, $g(x)$ any positive function)? Thanks.

Answer 1

Let $h(x)=f(x)\cdot g(x)\forall x\in\Bbb R$

Now if $h(x_0)$ is $0$, i.e, $x_0$ is a root of $h(x)$,

$$f(x_0)\cdot g(x_0)=0$$

$$f(x_0)=0 \text{ or }g(x_0)=0$$

But we are given that $g(x_0)\ne0$

$\therefore f(x_0)=0$

Therefore $h(x')=0 \iff f(x')=0\implies$ $h$ and $f$ have the same roots.

Answer 2

Consider $fg=0$. Since $g\neq 0$ hence dividing by $g$ makes sense and therefore you'll get the roots of $fg$ to be same as roots of $f$.