If g(x) and f(x) are polynomial functions, does the rational function g(x)/f(x) mean g(x) ÷ f(x) ? So can g(x) / f(x) be evaluated by the long division of the polynomials, where g(x) is the divident and f(x) is the divisor?
So when the fraction bar symbol is used with polynomial expressions does it have the same meaning, as it does when it is used with numbers?
I have just started learning about rational functions, and I was a bit confused about that. I got a vague response from my teacher so I thought I would ask here.
You are correct that $\frac{g(x)}{f(x)}$ can be evaluated using polynomial long division.
Additionally, you can view any formula, including rational functions, as a series of instructions to apply to some value $x$. For instance, $\frac{x^2}{x+1}$ means to take a value $x$, square it, then divide by the original value plus one.
The key thing is that when you use polynomial long division, you get an equivalent formula (except when dividing by zero). For instance, $x^2$ divided by $x+1$ is $x-1$ remainder $1$, so $$\frac{x^2}{x+1} = x-1+\frac{1}{x+1}.$$ This is not just a true fact about polynomials, but also regardless of what values of $x$ you plug into both sides of the equation, you'll get the same number (except when dividing by zero).