1. Consider the function [math]f\left(x\right)=5x^3+4x-7[/math]. If we divided [math]f\left(x\right)[/math] by [math]x-3[/math], what would you expect the degree of the remainder, [math]r_1[/math], to be? What do you expect the graph of the remainder to look like?[br]
2. Let [math]r_2[/math] be the remainder when we divide [math]f\left(x\right)[/math] by [math]\left(x-3\right)^2[/math], and let [math]r_3[/math] be the remainder when we divide [math]f\left(x\right)[/math] by [math]\left(x-3\right)^3[/math]. What do you expect the degree of [math]r_2[/math] and the degree of [math]r_3[/math] to be? What expectations (if any) do you have about the shape of the graphs of these remainders?
3. Graph[math]f\left(x\right)[/math], [math]r_1[/math], [math]r_2[/math], and [math]r_3[/math] on the set of axes below.
4. Did the graphs of the three remainders meet your expectations? What do [math]r_1[/math], [math]r_2[/math], and [math]r_3[/math] all have in common?
5. If we let [math]f\left(x\right)[/math] be an arbitrary polynomial of degree [math]n[/math], can you make a conjecture about the remainder [math]r[/math] when we divide [math]f\left(x\right)[/math] by [math]\left(x-a\right)^m[/math] with [math]n\ge m[/math]?