# What is a polynomial and how is it different from a function?

I have a problem that asks me to find a polynomial $P(x)$ so that $P(3)$ is 9.

Now I can say with certainty that $P(x)$ can be $x^2$. This is a second degree polynomial.

But what about functions such as $\frac{1}{x^2+3}$, are these not polynomials? If not, why?

P.S. I seem to be having troubles with the math tags. Some help on that would be appreciated as well.

• Use dollar signs either side of mathematics. For more information, see the guides here. Oct 31 '13 at 21:44
• You could use the polynomial $P(x) = 9$, or $P(x) = 9+(x-3)^{1001}$. Oct 31 '13 at 21:45
• Polynomials in an indeterminate $x$ (normally) don't have terms of negative powers of $x$. They're defined that way. Functions do. Oct 31 '13 at 21:46
• Also $P(x)=3x$ will do, or $P(x)=4x-3$. There are infinitely many polynomials with the property, in any (positive) degree. Oct 31 '13 at 21:47
• @Shaun Functions don't have to have negative powers at all. Oct 31 '13 at 22:16

A polynomial (in one variable, over the real numbers) is by definition an expression of the form

$f(x) = a_0 + a_1x+a_2x^2 + \cdots + a_nx^n$

for some non-negative integer $n$ and real numbers $a_i,\ i=0,1,\dotsc, n$.

Note that the variable $x$ must have non-negative powers; things like $r(x) = \frac{1}{x^2+3}$ are, by definition, not polynomials in $x$ since they include negative powers of $x$. (Properly speaking, it contains a negative power of the polynomial $x^2-3$, and hence cannot be written in the same form as $f(x)$ above.)

Of course, the numerator and denominator of $r$ are both polynomials; expressions of this form $\frac{p}{q}$ where $p,q$ are polynomials are called rational functions.

• Nice explanation! :) Oct 31 '13 at 21:52
• I don't really see a "negative power of $x$" in $1/(x^2+3)$. The only negative power is of $x^2+3$, not of $x$ itself. Oct 31 '13 at 21:53

Polynomials are finite formal sums $\sum_{i = 0}^n a_i x^i$, where $a_i$ belongs to some ring (if you need some idea of what a ring can be, the real numbers are a ring, the integers are a ring, and the rationals are a ring). By definition, there are no negative powers of the indeterminate $x$ involved (likewise, there are no reciprocals of expressions involving $x$).

Technically, a polynomial is not a function, although you can consider it to be one if you take an element $r$ in the original ring and "plug it in" (let $x = r$ and compute). However, if you interpret a polynomial as a function in that manner, two different polynomials can give the same function. For example, consider the field $\Bbb Z/p\Bbb Z$ (i.e. integers modulo $p$). The polynomials $0$ and $x^p - x$ define the same function by Fermat's little theorem, but they are not the same polynomial.

• While all of this is fine, the technical nature of this answer I fear is much too advanced for the level of question. Oct 31 '13 at 22:21
• Annoying precision got the better of me once again! :P Oct 31 '13 at 23:03
• To add to this, Oct 31 '13 at 23:39