Creating a Polynomial Function

One thing that I have been looking forward to trying in Advanced Algebra is more tasks. One of my goals for this year is doing more mathematical talking and orchestrating better discussions in the classroom. One task I created recently that still needs work is this creating a polynomial function from zeros task. Previously students have only factored and solved polynomials with P's and Q's. I was thinking they could work their way backwards from the previous days material.

Here is a link to a better looking version.

I gave them the task to work on for 15 minutes and I went around the classroom getting students started and answering questions.  If I had more time I would have done a better job of calling students up and presenting their work, but I was running short on time. I was hoping their thinking would help them think backwards especially with day before lesson.

Some places they might have gotten stuck at that I had posed responses for:

• Students will have a hard time getting started (it feels like a high entry task)
• I will ask them to go through an example problem we have done before, such as x^2-4x+3
• Multiplying the "i's" together, it has been about a full quarter since they have seen imaginary numbers multiplied together.
• I will ask them what i squared is? What do they remember about imaginary numbers?
• Distributing correctly.
• Ask them how they would distribute (x+2)(x+2)?

If you are really digging this lesson, you can view 28 minutes of the lesson here. 

I still feel like the entry to the task is too high, I would love comments on how this could be a better lesson.