Polynomials are best understood by creeping up on them. Take a look at the expression
It consists of a variable which is multiplied by . The constant is then added on. This an example of a linear expression. If you treat it as a mathematical function and plot its graph, you get a straight line (see the figure below). The general form of such a linear expression is
where and are two constants.
A quadratic expression is of the form
where , , and are constants. The difference to a linear expression is that we've added on a multiple of the square of . An example of a quadratic expression is
Here we have and If you treat this expression as a mathematical function and plot its graph, you get a parabola (see the figure below).
We don't need to stop here. We could also add on a multiple of the cube of to get the general cubic expression
and then add on a multiple of to the fourth power to get the general quartic expression
Taking this further, a polynomial in the variable is any expression formed by taking multiples of the powers of and adding them together (note that , so even the last term can be viewed as containing a power of ). We can write a general polynomial as
where is the highest power of that appears in the expression and , , etc, up to are constants. These constants are called the coefficients of in the polynomial.
The word "polynomial" captures the fact that a polynomial is made up of many terms. It's a mixture of Greek and Latin, derived from the Greek word "poly", which means "many", and the Latin word "nomen", which means "name", but in this case can be interpreted as meaning "term".

The blue line is defined by the polynomial function f(x)=2x+1, the orange parabola by f(x)=x2, the green curve by f(x)=x3+3x2, and the red curve by f(x)=x4+4x3+6x2+4x+1.
Specify and generalise
There are a couple more things we need to say to make our definition of a polynomial complete. Note that in the above we haven't specified what kind of number the variable represents, and what kind of values the coefficients can take.
When you meet polynomials at school, you usually assume that the coefficients, as well as the variable, are all real numbers. There might be situations, however, where you want the variable and/or coefficients to only take integer or fractional values, or where you may allow them to take on complex values. Generally, when talking about polynomials, you should specify the sets of numbers the coefficients and the variable belong to. For example, you might talk about a polynomial in a real variable with integer coefficients.
The second thing to note is that you can also have polynomial expressions in more than one variable. The expression
is an example of a polynomial with two variables and the expression
is an example of a polynomial with three variables. In general, a polynomial in several variables is a sum of many terms, each of which is the product of powers of these variables multiplied by a coefficient.
Geometry from algebra
Finally, we should point out the link between polynomials and geometry. As we've seen above, polynomials in one variable can define lines and curves. The link comes about through the notion of a coordinate system. Similarly, a polynomial in two variables can define a surface. For example, the surface shown below is defined by the polynomial function . Polynomials with more than two variables can also be thought of as defining geometric objects, though in this case the objects live in higher dimensions we can no longer visualise. See Maths in a minute: Algebraic varieties to find out more.

This surface is defined in a three-dimensional coordinate system by the equation f(x,y)=x3+y3+1.
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