$(5x +1) ÷ (3x)$ is not a polynomial?

Question

On the Mathwarehouse page on polynomial equations, it gives this expression as an antiexample, something that is not a polynomial: $(5x +1) ÷ (3x)$

However, it also says on the same page that if it is possible to simplify an expression using addition, subtraction or multiplication, then it is a polynomial.

Following that logic, it would seem to me that their anti-example is actually a polynomial, as it seems that it can be rewritten as: $(5x + 1)(\frac 13 x)$

So which is correct? Is it or is it not a polynomial?

Answer 1

It is not a polynomial. Your simplification does not work: the first expression is $$ \frac{5x+1}{3x}. $$ Your second expression has $x$ and $1/x$ mixed up, basically.

Answer 2

It is not a polynomial.

A polynomial can NOT include a variable with negative power.

Answer 3

Definition: $f(x)$ is a real polynomial if it can be written on the form $f(x) = \sum_{k=0}^n a_k x^k = a_0 + a_1x + \ldots + a_n x^n$ where $n$ is an integer and $a_k$ are real numbers.

From this definition it is not hard to prove the following:

If $f$ and $g$ are polynomials then $af(x)+bg(x)$ for $a,b\in\mathbb{R}$ and $f(x)g(x)$ are also polynomials

However polynomials are not closed under division. Dividing two polynomials can give a polynomial, but it does not have to and your example is not a polynomial.

One way to prove this is to assume that $\frac{5x+1}{3x}$ is a polynomial and try to derive a contradiction. By the definition above we can write $\frac{5x+1}{3x} = a_0 + a_1x + \ldots +a_n x^n$. By considering the growth of the two sides of the equation as $x\to \infty$ we conclude that we must have $a_1,a_2,\ldots,a_n=0$. This leaves us with $\frac{5x+1}{3x} = a_0$ a constant. This is absurd and it follows that $\frac{5x+1}{3x}$ cannot be a polynomial.