Polynomials

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Let’s talk about some of the nicest functions you’ll ever have the pleasure of dealing with: the polynomials.  These are functions which are sums of numbers times integer powers of a variable, for example 4x+3, t^2+t+1, and 17x^{22}-9x^{18}+9x^{15}-x-1.  Functions that have negative powers of a variable, like x+x^{-1}, and functions that have non-integer powers of a variable, like x^{3/2}, are not polynomials.  Other functions such as \sin t, \ln (x-1), and e^{2x}, are also not polynomials.

As I said above, polynomials are very nice functions.  Their domain is all of the real numbers, and they’re continuous everywhere.  This means, for one thing, that it’s very easy to calculate limits of polynomials: just plug in the limiting value!  For example, to find the following limit we just plug in two:

\lim_{x\rightarrow 2} x^3-x^2+x-1 = 2^3-2^2+2-1

\quad\quad = 5

The highest power of the variable in a polynomail function is called the degree of the polynomial.  The degree zero polynomials are just constant functions.  You probably already know the degree one polynomials, since they’re functions like x+1 and -2x+3.  These are the linear functions.   Degree two functions are called quadratic functions, and their graphs look like parabolas.

Here are a few of questions to ask yourself:

  • What do the graphs of degree three polynomials look like?  Degree four polynomails?  Howabout five?  Is there a pattern?
  • What kind of function is the derivative of a polynomial?
  • If f(x) is a polynomial, what can you say about \lim_{x\rightarrow\infty} f(x) and \lim_{x\rightarrow -\infty}?