CAS polyRemainder( , and related commands, are used to explore the relationship between the remainder when f (x) is divided by the linear factor (x-a), and f (a). This involves calculating remainders when dividing f(x) by (x-a) for captured values of "a" using CAS, and scatterplotting those values against the values of the function, f(a) in real time. The visualization of plotting remainders against function values "live" illustrates the symbolism in a new and concrete way.
Before the Activity
Students should understand long division and/or synthetic division as well as the Fundamental Theorem of Arithmetic.
During the Activity
During the activity you may need to note that the polyRemainder( command does not easily lend itself to working with lists, multiple inputs of any kind, or data captures. Hence, the formulae seen in the spreadsheet, were inserted in b1 and filled downward to b10.
After the Activity
Various questioning types lend themselves to assessing competence on this content such as multiple choice, short answer, graphical interpretation, etc. It would be encouraging and interesting to direct a question as indicated on pages 4.3 or 4.4. For example, f(x) divided by (x-1)is positive. Therefore, which is true about f(1)? (A) It is positive (B) It is negative (C) It is zero (D) No conclusion can be drawn.