Polynomial Functions

This page contains an applet to help you explore polynomials of degrees up to 5:

f(x) = ax5 + bx4 + cx3 + dx2 + ex + f



by changing the values of the coefficients a, b, c, d, e and f. It is not easy to draw any conclusion when you change all 5 coefficients at the same time. You can always reduce the degree (highest power) by setting some parameters to zero. For example if you set parameter a to zero and b to a non zero value, you obtain a polynomial of degree 4. Once you finish this interactive tutorial, you may want to consider a Graphs of polynomial functions - Questions. If needed, Free graph paper is available.

Interactive Tutorial Using Java Applet



A polynomial f(x) with real coefficients and of degree n has n zeros (not necessarily all different). Some or all are real zeros and appear as x-intercepts when f(x) is graphed.

A - Explore real solutions of polynomial equations of the form

xn + f = 0


where n is even or odd and f is a constant.



Click the button "start here" to start the applet and maximize the window obtained.

  1. Use the scrollbar to set all coefficients to zero except a and f. Write down the polynomial and its degree, examine the graph obtained. How many x-intercepts ( or real solutions to the above equation ) the graph has? Repeat for different values of a and f

  2. Use the scrollbar to set all coefficients to zero except b and f. Write down the polynomial and its degree, examine the graph obtained. Change b and f and see how many x-intercepts the graph has? Which values of f give intercepts and which values do not give any intercepts?

  3. Use the scrollbar to set all coefficients to zero except c and f. Write down the polynomial and its degree, examine the graph you obtain. Change c and f and see how many x-intercepts the graph has?

  4. Use the scrollbar to set all coefficients to zero except d an f. Write down the polynomial and its degree, examine the graph you obtain. Change d and c and see how many x-intercepts the graph has and for what values of f

  5. Use the scrollbar to set all coefficients to zero except e and f, write down the polynomial and its degree, examine the graph you obtain. Change e and f and see how many x-intercepts the graph has?

    From 1,2,3,4, and 5 above, what conclusion can you make as to the number of solutions of polynomials equations of the form
    xn + f = 0

    depending on whether n is even or odd and f is negative, positive or zero?.



B - Explore even and odd polynomials


  1. Use the scrollbar to set a, c and e to zero, write down the polynomial and its degree, examine the graph you obtain, is f(x) even, odd or neither?

  2. Use the scrollbar to set b,d and f to zero, write down the polynomial and its degree, examine the graph you obtain, is f(x) even, odd or neither?

C - More on the zeros of polynomials


NOTE: For the next tutorial, a repeated zero of multiplicity m is counted m times.

  1. Set a to a non zero value (polynomial of degree 5). Change all the other coefficients (non zero values if possible) so that the graph of f(x) has:
    1 x-intercept.
    3 x-intercepts.
    5 x-intercepts.
    Why do you think we can obtain only an odd number of real zeros of f(x)?

  2. Set a to zero and b to a non zero value (polynomial of degree 4). Change all the other coefficients (non zero values if possible) so that the graph of f(x) has:
  3. no x-intercept.
  4. 2 x-intercepts.
  5. 4 x-intercepts.
    Why do you think we can obtain only an even number of real zeros for f(x)?

    (The last two exercises are not easy, however they are very educative about polynomials).

D - Leading Coefficient Test


  1. Set parameter a (leading coefficient) to a positive value (polynomial of degree 5) and set b, c, d, e and f to some values.
    As x increases without bounds, does the right side of the graph rise or fall?
    As x decreases without bounds, does the right side of the graph rise or fall?
    Change b, c, d, e and f and see if the above behavior changes.

  2. Set a to zero and b (leading coefficient) to a positive value (polynomial of degree 4) and do the same exploration as in 1 above and 2 above.

  3. Set a and b to zero and c (leading coefficient) to a positive value (polynomial of degree 3) and do the same exploration as in 1 above and 2 above.

  4. Set a, b and c to zero and d (leading coefficient) to a positive value (polynomial of degree 2) and do the same exploration as in 1 above and 2 above.

  5. Set a, b, c and d to zero and e (leading coefficient) to a positive value (polynomial of degree 1) and do the same exploration as in 1 above and 2 above.

    What can you say about the behavior of the graph of the polynomial f(x) with an even degree n and a positive leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

    What can you say about the behavior of the graph of the polynomial f(x) with an even degree n and a negative leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

    What can you say about the behavior of the graph of the polynomial f(x) with a odd degree n and a positive leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

    What can you say about the behavior of the graph of the polynomial f(x) with a odd degree n and a negative leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?



More references and links to polynomial functions.
Derivatives of Polynomial Functions.

Polynomial Functions, Zeros, Factors and Intercepts

Find Zeros of Polynomial Functions - Problems

Multiplicity of Zeros and Graphs Polynomials.

Graphs of Polynomial Functions - Questions.


Find the Inverse Functions - Online Calculator
Free Online Graph Plotter for All Devices
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Updated: 2 April 2013

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