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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 tex2html_wrap_inline304 is a polynomial while the function tex2html_wrap_inline306 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:

    displaymath34

    In this form, n is the degree and tex2html_wrap_inline310 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 tex2html_wrap_inline318 is the constant function. Its graph is a horizontal line passing through the points where tex2html_wrap_inline320 .
  2. The function tex2html_wrap_inline322 is a straight line with slope tex2html_wrap_inline324 and y-intercept tex2html_wrap_inline328 .
  3. The function tex2html_wrap_inline330 is a parabola. By completing the square, we can rewrite the function as tex2html_wrap_inline332 where tex2html_wrap_inline334 and tex2html_wrap_inline336 . 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 tex2html_wrap_inline310 .

  1. If n is odd, then the behavior of the function (increasing or decreasing) as tex2html_wrap_inline348 and the behavior as tex2html_wrap_inline350 are opposite.
    1. If tex2html_wrap_inline352 then the function ``rises to the right'' (ie. increases as x increases.)
    2. If tex2html_wrap_inline356 then the function ``falls to the right'' (ie. decreases as x increases.)
  2. If n is even, then the behavior of the function as tex2html_wrap_inline362 is the same.
    1. If tex2html_wrap_inline352 then the function rises to the right (and left.)
    2. If tex2html_wrap_inline356 then the function falls to the right (and left.)

next up previous
Next: Exponentials Up: Some Important Functions Previous: Some Important Functions

Vector Calculus
Wed Sep 17 14:50:13 MST 1997