Multiplicity of Zeros and Graphs Polynomials

An app is used to explore the effects of multiplicities of zeros and the leading coefficient on the graphs of polynomials the form: \[ f(x) = a(x-z_1)(x-z_2)(x-z_3)(x-z_4)(x-z_5) \]

With this factored form, you can change the values of the leading coefficient a and the 5 zeros \( z_1, z_2, z_3, z_4 \) and \( z_5 \). You can explore the local behavior of the graphs of these polynomials near zeros with multiplicity greater than 1.
Once you finish this interactive tutorial, you may want to consider a self test on graphs of polynomial functions.

Interactive Tutorial on Zeros of Polynomials

Click the button "Plot Polynomial" to start.
You may hover the mousse cursor over the graph to trace the coordinates. You may also Hover the mousse cursor on the top right of the graph to have the option to download the graph as a png file.
Some activities that you may wish to go through are listed below.

\(a \) = \(z_1 \) = \(z_2 \) = \(z_3 \) = \(z_4 \) = \(z_5 \) =


Activities
When you open this page, the default value of the leading coefficient \( a \) is 1 and those of the the zeros are \(z_1 = -1 \), \(z_2 = 0 \), \(z_3 = 2 \), \(z_4 = 1 \) and \(z_5 = 3 \). Note that the zeros are represented graphically by x intercepts.

1 - Set the values of \( z_1,z_2,z_3,z_4 \) and \( z_5 \) to zero, then change the value of the leading coefficient \( a \).
How does coefficient \( a \) affect the behaviour of the graph of the polynomial \( f(x) \) as x increases indefinitely (right side behaviour) and as x decreaseses indefinitely (left side behaviour)?
Change \( a \) from a positive to negative values and note the effects it has on the graph.

2 - Set coefficient \( a \) to a certain value (not zero) and set \( z_1,z_2,z_3,z_4 \) and \( z_5 \) to the same value, 2 for example.
How does this choice affect the graph of f(x)? Write down the equation of f(x).

3 - Set \( z_1 \) and \( z_2 \) to the same value (say -2); this is a zero of multiplicity 2. Set \( z_3, z_4\) and \( z_5 \) to another same value (say 1); this is a zero of multiplicity 3.
Write down the equation of f(x).
What is the shape of the graph locally, around the zeros, at -2 (mulitplicity 2) and 1 (mulitplicity 3)?.
How does the multiplicity of the zeros affect the graph locally around the zeros. For which multiplicity does the graph cut or touch the x axis?

4 - Set \( z_1 \) to a value (say 3); this is a zero of multiplicity 1. Set \( z_2, z_3, z_4\) and \( z_5 \) to another same value (say 2); this is a zero of multiplicity 4.
Write down the equation of f(x).
What is the shape of the graph locally, around the zeros, at 3 (mulitplicity 1) and 2 (mulitplicity 4) ?. For which multiplicity does the graph cut or touch the x axis?

Conclusion
Does the graph of a polynomial cut or touch the x axis at an even multiplicity such as 2, 4, ...
Does the graph of a polynomial cut or touch the x axis at an odd multiplicity such as 1, 3, 5, ...


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More references and links to polynomial functions

Derivatives of Polynomial Functions.
Polynomial Functions
Polynomial Functions, Zeros, Factors and Intercepts
Find Zeros of Polynomial Functions - ProblemsGraphs of Polynomial Functions - Questions.
Factor Polynomials.