A large screen applet helps you explore the effects of multiplicities of zeros on the graphs of polynomials the form:
f(x) = a(x-z1)(x-z2)(x-z3)(x-z4)(x-z5)
With this factored form, you can change the values of the leading coefficient a and the 5 zeros z1, z2, z3, z4 and z5. 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.
How do the leading coefficient a and the zeros z1,z2,z3,z4 and z5
affect the graph of f(x)?
1- Use the scrollbar to set z1,z2,z3,z4 and z5 to zero, then
change the value of a.
How does a affect the graph of f(x)? Change a from a positive
value a1 to a negative
value -a1 and note the effects it has on the graph.
2- Set a to a certain value (not zero) and set z1,z2,z3,z4
and z5 to the same value z.
How does this choice affect the graph of f(x)? Write down the
equation of f(x).
3- Set z1 and z2 to the same value (say z11) and z3,z4 and z5 to another and same value (say z22). Write down the equation of f(x). What is the
shape of the graph locally (around the zeros) at z11 and z22 ?.
How does the multiplicity of the zeros affect the graph locally (around the zeros).
More references and links to polynomial functions.