Polynomials

Polynomials are that class of functions that are made up of several terms, each of which is a combination of real-valued coefficients and integral powers of the independent variable. Thus, the function is a polynomial while the function is not. Some important things to recall about polynomials are listed below.

1. Polynomials are continuous for all real numbers.
2. The highest integral power appearing in a polynomial is called the order or degree of the polynomial.
3. It is usual to write a polynomial in standard form:

   

   In this form, n is the degree and is the leading coefficient.

4. An nth degree polynomial has at most n real roots (zeros) and n-1 turning points, where the graph changes behavior from increasing to decreasing or vice versa.
5. The graph of a polynomial is always smooth, containing no kinks bends or asymptotes.

Some of the most common types of polynomials are listed here.

1. The polynomial is the constant function. Its graph is a horizontal line passing through the points where .
2. The function is a straight line with slope and y-intercept .
3. The function is a parabola. By completing the square, we can rewrite the function as where and . The vertex of the parabola is at the point (h,k) and the parabola opens upward if a > 0, downward if a < 0.

When graphing polynomials, it is usually a good idea to start with locating the zeros. Since complex roots come in conjugate pairs, we know that odd degree polynomials must have at least one real root. Often we must use a combination of factoring, long division (or synthetic division), the quadratic formula, and numerical techniques to locate the zeros.

It is also helpful to note the following when graphing a polynomial of degree n and having leading coefficient .

1. If n is odd, then the behavior of the function (increasing or decreasing) as and the behavior as are opposite.
   1. If then the function ``rises to the right'' (ie. increases as x increases.)
   2. If then the function ``falls to the right'' (ie. decreases as x increases.)
2. If n is even, then the behavior of the function as is the same.
   1. If then the function rises to the right (and left.)
   2. If then the function falls to the right (and left.)