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Is f(x)=(8x^3)-(4x^2)+(2/x) a polynomial function? I wasn't sure because couldn't 2/x be written as 2×(x^-1), which would make it not a function?

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    Can you tell us your textbook or instructor gives as the definition for a polynomial? – 2017-01-30
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    No i was helping a friend and neither of us knew what it was. I have a harder book than her, so I don't have the specific book definition. – 2017-01-30
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    I see. FYI, polynomials, as you seem to have guessed, usually only have positive integer powers on the variables. – 2017-01-30

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It's what's called a rational function: fraction of two polynomials.

The function can be rewritten: $$f(x)=\frac{8x^4-4x^3+2}{x},$$ and this has a polynomial as both numerator and denominator and hence it's a rational function.

All polynomials are rational functions since if $P(x)$ is a polynomial then it can be written as $P(x)/1$ where $1$ is seen as a constant polynomial. However obviously, not all rational functions are polynomials.

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    The OP also needs a *definition* of a polynomial function, e.g. is $\,x/x\,$ a polynomial function? – 2017-01-30
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    a polynomial in the real/complex field is a function of the form $a_n x^n+a_{n-1}x^{n-1}+...+a_1 x+a_0$ where $n$ is a nonnegative integer and $a_0,...,a_n\in\mathbb{R}$ / $\mathbb{C}$. $x/x$, just the way it's given, isn't defined at $x=0$ but everywhere else it's equal to the constant polynomial $1$. If we define a function to be $x/x$ for all $x\neq 0$ and equal to $1$ at $x=0$, then we actually do just get the 1 polynomial. – 2017-01-30
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    The correct answer is: it depends on the context. For example, in algebra, in the field of rational functions we have the equality $\,x/x = 1\,$ is a constant polynomial function. In analysis this may or may not be true depending on the context. So there are some subtleties. – 2017-01-30