Lecture 17

Learning objectives

After this class, you should be able to:

1. Prove the theorem on page 340.
2. Apply the above theorem to express a rational function as partial fractions, and to solve a recurrence using generating functions.

Reading assignment

1. Page 338-340.
2. Review division on polynomials.

Exercises and review questions

• Questions on current lecture's material
  1. Use generating functions to derive the closed for solution to the following recurrence: g0 = 1/4, g1 = 1/4, gn = (3/2) gn-1 - (1/2) gn-2.
  2. Verify the correctness of the partial fraction expansion that we performed for the following rational function: z/(z2 - 2z + 2).

• Questions on next lecture's material
  1. Find the integer and fractional parts for the following division: 1000/11.
  2. Express the following rational function as T(z) + P(z)/Q(z), where P(z) is a lower degree polynomial than Q(z), and T(z) is some polynomial: z3/(z+1).
  3. Express the following rational function as T(z) + P(z)/Q(z), where P(z) is a lower degree polynomial than Q(z), and T(z) is some polynomial: z3/(z-1).
  4. Express the following rational function as T(z) + P(z)/Q(z), where P(z) is a lower degree polynomial than Q(z), and T(z) is some polynomial: (2 z3 + 2 z2 - 2 z - 2)/(z-1).