1. 7.4 Partial Fraction Decomposition

3. A rational expression P / Q is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. Otherwise the rational expression is termed improper.

4. CASE 1: Q has only non-repeated linear factors. Under the assumption that Q has only non-repeated linear factors, the polynomial Q has the form where none of the numbers a i are equal. In this case, the partial fraction decomposition of P / Q is of the form where the numbers A i are to be determined.

5. Write a partial fraction decomposition of

6. Solution of this system is A=3, B=2.

7. CASE 2: Q has repeated linear factors. If the polynomial Q has a repeated factor, say, (x - a) n, n > 2 an integer, then, in the partial fraction decomposition of P / Q, we allow for the terms

8. Write the partial fraction decomposition of

9. From the last and the first we easily get A and B. A=-3/4, B=3/4, C=7/2. Need to solve:

11. CASE 3: Q contains a non-repeated irreducible quadratic factor. If Q contains a non-repeated irreducible quadratic factor of the form ax 2 + bx + c, then, in the partial fraction decomposition of P / Q, allow for the term where the numbers A and B are to be determined.

12. Write a partial fraction decomposition of =

13. = Collect like terms: Comparing right and left sides gives:

14. Adding (1) and (3) gives 2A+2B = 4 Adding (2) and (4) gives 2A-2B = -8 Combining these 4A = -4 (1) (2) (3) (4) Solution is A = -1, B = 3, C = 2, D = 1.

15. CASE 4: Q contains repeated irreducible quadratic factors. If the polynomial Q contains a repeated irreducible quadratic fator (ax 2 + bx + c) n, n > 2, n an integer, then, in the partial fraction decomposition of P / Q, allow for the terms where the numbers A i,B i are to be determined.

16. Write the partial fraction decomposition of:

17. Comparing left and right sides leads to: