Determining if polynomial function despite no addition/subtraction or rasing to power

Question

f1(n) = 100n

Is the above function polynomial? I have examples of polynomial fucntions, but this one particularly I am not sure about since there is no addition or subtraction being made afterwards. Also, it is not at any time being raised to the power of another number so it seems very odd to call it polynomial.

I'm after an explanation or 'why' as opposed to yes/no

Answer 1

You are right to ask, because this is really a question of language and what people have chosen that the words should mean. And that, in turn, depends on what definition is interesting.

For the specific question of polynomials, a polynomial is a sum of terms each of which is a constant multiplied by a power.

There have to be at least zero terms in the sum. That’s fine. In your case there is one.

The powers have to be at least zero. That’s fine. In your case you have $n$ raised to the 1st power.

Sometimes mathematicians assume additional criteria and don’t explicitly specify them because they think they are obvious. This is annoying but you hust have to get used to it. For instance: is $100n$ a quadratic polynomial - that is, of the form $ax^2+bx+c$?

• obviously yes, with $a=0$ and $c=0$.
• obviously no, because the standard formula for the solution of a quadratic equation doesn’t work, and statements like “every quadratic has two solutions” can only be made to work with unnatural qualifications such as “and one of them is infinite”.

To summarise: “something is a polynomial” is not a fact but a linguistic decision, and in your particular case the answer is “yes, it is”.

Answer 2

A polynomial function is a function $f:\mathbb R\to\mathbb R$ for which there are real numbers $a_0,\dots,a_n$ such that $$ f(x)=\sum_{i=0}^{n}a_ix^i=a_0+a_1x+\dots+a_nx^n \, . $$ This means that every function of the form $f(x)=a_0+a_1x$, i.e. every linear function, is a polynomial function. Your function is a linear function (with $a_0=0$ and $a_1=100$), and hence is a polynomial function. A polynomial does not have to be a sum of two or more terms, and the highest power of $x$ is allowed to be $1$ (or $0$ for that matter). Therefore, the following are all examples of polynomials: \begin{align} f(x)&=x+2 \\[4pt] g(x)&=x^2 \\[4pt] h(x)&=0 \, . \end{align}