Polynomials and their roots

What are the polynomials? They are a particular collection of functions.

The idea of polynomials

Polynomials are the (smallest) collection of functions satisfying:

  • The constant functions and the identity function f(x)=x are polynomials.
  • If you combine two polynomials using the operations addition, subtraction, or multiplication, then the result is a polynomial.

The definition of polynomials

Definition: a polynomial function is any function of the form f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} for fixed numbers a_{n}, \ldots, a_{0} \left(a_{n} \neq 0\right) and n is called its degree.

Theorem. If a polynomial f(x) has a root r, then f(x) can be written in the form f(x)=(x-r)\cdot q(x), where q(x) has degree one less than the degree of f(x).

What does this theorem allow us to say about the roots of a polynomial f(x) of degree 4? Discuss

Theorem. If a polynomial f(x) has odd degree, then it has at least one root.

Think about:

  • Can all polynomials be factored?
  • Does every polynomial have a root?
  • Is it possible to factor a polynomial that has no root? (ex: x^4+3x^2+2)

Rational Functions

The interest in rational functions, as far as school mathematics is concerned, is twofold: they are only one step away from polynomials and are therefore worth knowing, and surprisingly, their graphs display features that are genuinely different from those of polynomials—they have asymptotes—and are therefore instructive for that reason.

Wu, A&G, p131

Indeed, rational functions provide a great way to introduce many new and interesting properties of functions. These include:

  • vertical asymptotes
  • horizontal asymptotes
  • holes in the graph
  • restricted domains (values of x for which the function is undefined)

The idea of rational functions:
“Just like polynomials but division is allowed”

Rational functions are the (smallest) collection of functions satisfying:

  • The constant functions and the identity function f(x)=x are rational functions.
  • If you combine two rational functions using the operations addition, subtraction, multiplication, or division, then the result is a rational function.

The definition of rational functions

Definition: a rational function is any function of the form f(x)=\frac{p(x)}{q(x)} where p(x) and q(x) are polynomials. The domain consists of all real numbers x for which q(x)\neq 0.