I understand polynomials in one variable as an algebraic expression that is made up of many terms, which consists of coefficients for example $a_n\dots a_0$ that are real numbers.

$$a_n x^n+ a_{n-1} x^{n-1}+ … + a_2 x^2+ a_1 x+ a_0$$

An example is:

$$2x^4 - 7.5x^3 + x - 12.$$

what I don't get is how does the expression of polynomial relate to my example? what does $a_n$, $a_{n-1}$ mean or what does the expression even mean?

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The coefficients do not have to be real numbers. They could be complex numbers or elements of certain more general algebraic systems (e.g., any commutative ring). – KCd Oct 8 '11 at 16:41
$a_4=2,a_3=-7.5,a_2=0,a_1=1,a_0=-12$ – Américo Tavares Oct 8 '11 at 16:41
I see, from $a_2 = 0$ its starting to make sense now.. – John Oct 8 '11 at 16:58
One usual convention is to index coefficients by the exponent of the variable they're multiplied with, e.g. $a_3$ for the term multiplying $x^3$. There is also a "reversed" convention where the polynomial looks like $a_0 x^n + a_1 x^{n-1} + \cdots + a_n$. It depends on the author. – J. M. Oct 8 '11 at 17:08

An expression of the form $$a_nx^n+ a_{n-1}x^{n-1}+\ldots +a_1 x+a_0$$ can be interpreted in two ways:

(a) The $a_k$ $(0\leq k\leq n)$ are certain constants taken from a "ground field" $R$, e.g., $R:={\mathbb R}$, and $x$ is a variable taking values in $R$, so it makes sense to compute powers $x^k$. All in all such an expression defines a function $$p:\quad R\to R,\quad x\mapsto p(x):=a_nx^n+ a_{n-1}x^{n-1}+\ldots +a_1 x+a_0\ .$$

Example: If $n=2$ and $a_0=3$, $a_1=0$ and $a_2=7$ then $p(x)=7x^2+3$. If $R={\mathbb R}$ then one can graph the function $x\mapsto p(x)$ in the usual way.

(b) The $a_k$ $(0\leq k\leq n)$ are certain constants taken from a "ground field" $K$, $X$ is a "variable taken from nowhere", called an "indeterminate", and one considers the "formal polynomial" $$p(X):=a_n X^n+ a_{n-1} X^{n-1}+\ldots +a_1 X+a_0\ ,$$ a purely algebraic entity which is completely determined by the coefficient vector $(a_0,a_1,\ldots, a_n)$. A standard task is then to construct a larger field $L\supseteq K$ such that the polynomial $p(X)$ factors as $$p(X)=a_n\ \prod_{k=1}^n(X-\lambda_k)$$ for certain $\lambda_k\in L$.

Concerning calculus the following is important: A given (real or complex) coefficient vector $(a_0,a_1,\ldots, a_n)$ defines the corresponding function $x\mapsto p(x)$ uniquely. But the converse is also true: A given polynomial function $p(x):=a_nx^n+ a_{n-1}x^{n-1}+\ldots +a_1 x+a_0$ cannot be realized by some other coefficient vector $(a_0',a_1',\ldots, a_m')$. The reason for this is that one has $$a_k={p^{(k)}(0)\over k!}\qquad(k\geq0)\ .$$

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Is it $L \supset K$ or $L \supseteq K$? – user2468 Oct 8 '11 at 18:55
@J.D.: Thank you; I have corrected this. – Christian Blatter Oct 8 '11 at 20:10