Polynomial FunctionsSignificancePolynomial functions are relatively easy to understand. Low degree polynomial equations can be solved explicitly. Polynomials provide good examples for studying more general functions.
Standard FormsThe standard form of a polynomial isThe a_{i} are real numbers and are called coefficients. The term a_{n} is assumed to be nonzero and is called the leading term. The degree of the polynomial is the largest exponent of x which appears in the polynomial  it is also the subscript on the leading term. A polynomial with one term is called a monomial. A degree 0 polynomial is a constant. A degree 1 polynomial is a linear function, a degree 2 polynomial is a quadratic function, a degree 3 polynomial a cubic, a degree 4 a quartic, and so on. Large Scale BehaviorThe behavior of a polynomial functionf(x) = a_{n}x^{n} + a_{n1}x^{n1} + ^{. . .} a_{1}x + a_{0} is essentially the same as that of the monomial over a large enough scale. To illustrate, this
consider the two functions
Exercise: Graph several polynomial functions together with their leading terms. Here are a few examples:
Zoom out on these graphs and watch the graphs of the polynomials and the graphs of the leading terms coalesce. Graphs of MonomialsThe general shape of the graph of a monomialdepends only on
As n gets larger, all graphs become
increasingly flat on the interval
Roots or Zeroes
A degree 2 polynomial is called a quadratic polynomial and can be written in the form
Its graph is a parabola. The quadratic formula states that
the roots of Roots of a polynomial can also be found if you can factor the polynomial. Examples and calculations are on the next page.
