A Cubic Polynomial

1. Consider the function . If we divided by , what would you expect the degree of the remainder, , to be? What do you expect the graph of the remainder to look like?

2. Let be the remainder when we divide by , and let be the remainder when we divide by . What do you expect the degree of and the degree of to be? What expectations (if any) do you have about the shape of the graphs of these remainders?

4. Did the graphs of the three remainders meet your expectations? What do , , and all have in common?

5. If we let be an arbitrary polynomial of degree , can you make a conjecture about the remainder when we divide by with ?