What's the value of a quadratic function when $x=0$ if the degree of the constant term is $0$?

Question

The degree of the constant term in a quadratic (or any polynomial) is $0$. Say, I have the following quadratic function: $$f(x) = 2x^2 + 4x +7$$

Since the degree of the constant term is $0$, I can also write the quadratic function like such: $$f(x) = 2x^2 + 4x^1 + 7x^0$$

If I plug in $x = 0$ into the function, I'm supposed to get the constant term. $$f(0) = 2(0)^2 + 4(0)^1 + 7(0)^0$$

But $0^0$ isn't defined. So does that mean that the $f(0)$ isn't defined? But if I didn't treat the constant as being multiplied by $x^0$, then I'll have an answer which equals $7$: $$f(0) = 2(0)^2 + 4(0)^1 + 7$$ $$f(0) = 7$$

I'm still learning about quadratics so I'd really appreciate someone clear me up about this without going into some advanced stuff.

Answer 1

We usually avoid using notation that will result in us having to deal with $0^0$ terms for exactly the reason you're describing, and hence we usually write polynomials as $p(x) = a_n x^n + \ldots + a_1 x + a_0$. However, we may also write the polynomial as $p(x) = \sum_{i = 0}^n a_i x^i$, which clearly does include an $x^0$ term. So what gives?

Broadly speaking, when it is clear that the assumption of $x^0 = 1$ even when $x = 0$ won't break anything, then it's ok to use that kind of shorthand. If needed, we can justify that we "won't break anything" by noting that this notation is 100% correct for $x \neq 0$, and also that it all still behaves nicely in the limit as $x$ approaches 0, i.e. $\lim_{x \rightarrow 1} x^0 = 1$ and hence defining $x^0 := 1$ even when $x = 0$ for this purpose results in continuity for all the polynomials, which is definitely a nice thing to have.

Of course, you can't use $0^0 = 1$ all the time - if we're looking at limits of the exponent, then $\lim_{y \rightarrow 0} 0^y = 0$ and so $\lim_{(x, y) \rightarrow (0, 0)} x^y$ is undefined. Thankfully, polynomials are only defined as having exponents that are non-negative integers, so that usually isn't something we have to worry about. If it is, then you might need to look at whether the $(0, 0)$ limit is going to be an issue and define your notation carefully to avoid it.