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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.
- 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
- 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?
- 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?
- 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
- 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
- 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?
- 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.
- 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)?
- 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:
- no x-intercept.
- 2 x-intercepts.
- 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
- 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.
- 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.
- 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.
- 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.
- 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.
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