I’m so excited for you to get to use this activity with your students.
You have 2 options available to you
1) Giving your students the equation and the picture printed on each page. There are dots on each picture to identify the whole number zeros. Students can use that information to divide out the real zero, and then get to a point where they can use the quadratic formula to solve for the remaining ones (some are going to be irrational, some will be imaginary).
2) Giving your students the equation only. These questions could turn out extremely lengthy if you have your students use the rational root theorem to list out all possible zeros, and check each option. You could opt to have your students use the graphing feature on their calculator to come up with the whole number zeros and continue on with division and the quadratic formula.
The answer key (with work) is attached.
Designed with purpose
There are 10 questions that get progressively harder (for a variety of reasons). Some include multiple whole number zeros, a zero with multiplicity of 2, a zero at zero, etc.
How I used this
I posted these pages around the room and explained to my students how the question #’s increased in difficulty. We had done an exit ticket the day before that was returned to them prior to this activity. I suggested that students who had gotten something wrong start with a question between #1 – 4. If a student got the exit ticket correct, that they start anywhere from #1 – 7. I suggested that from the previous day’s exit ticket where they were given 5 minutes to complete one of these questions, that with the time we had to do this activity (30 minutes) they should get through at least 6, if not more. Many of my advanced students got through all 10 in the 30-minute time frame, but 6 was a great bar to set for the time available.
Behind each question I posted the answer on a sticky note so students just had to lift the page to check their work. I did not post the answers where they could see and by looking across the room (like a scavenger hunt), because as soon as students saw that one real zero was at 5, and that there were 2 imaginary zeros, they would just have to see which answer had those characteristics without doing any of the practice.
I had the answer key in my possession with the work written out, so that if a student couldn’t figure out their mistake they could come consult with me. I was worried that students might just write down the answer and move on, but with the correct framing, I believe most every student took this as worthwhile time to practice.
Search Words: complex zeros, rational root theorem, rational zeros, quadratic formula, synthetic division, long division, polynomials, polynomial, imaginary numbers, irrational numbers