Quandaries
and Queries 

Who is asking: Student Level: Secondary Question: i also have to work out an equation that has the minimum possible degree for the properties to be simultaniously possible... and im stuck! 

Hi, I am going to do a similar problem.
Since there is a simple root at 1 the graph crosses the xaxis at x = 1. In the graph below I chose to start the graph below the xaxis and cross to above the xaxis at 1. The graph then has a double root at x = 2 so the graph comes back down to the xaxis and just touches it at x = 2, but doesn't cross the axis. Below is a sketch of a graph that meets these conditions.
If you want an equation of a polynomial that meets these requirements then the equation must have a factor of (x+1) if it has a root at 1 and it must have a factor of (x2)^{2} if it has a double root at 2. Since these ae the only conditions you have to satisify an equation with minimal degree would be
Suppose now that I change the problem slightly
One way to introduce a complex root is to put a dip in the graph, between x = 1 and x = 2, but a shallow enough dip that it doesn't cross the xaxis. To introduce this dip into the equation you can add a factor that is a quadratic with complex roots. I did it by adding the factor
so the equation becomes
This actually has two complex roots since complex roots come in pairs. Penny 

