8.2: Partial Fractions

8.2: Partial Fractions • Rational function • A ratio of two polynomials • Improper rational function • The degree of P is greater than or equal to the degree of Q. • Proper rational function • The degree of Q is greater than the degree of P.

Improper Rational Functions • Can be written as the sum of a polynomial and a proper rational function. • Where deg(P(x))  deg(Q(x)) and deg(r(x)) < deg(Q(x)) • Use long division of Q(x) into P(x) to accomplish this.

Examples

Partial Fractions Decomposition • A method for rewriting a proper rational function as a sum of simpler rational functions. • Let’s start with the proper rational function R(x) = P(x)/Q(x). • We need to consider 4 cases…

Case 1 • Q(x) factors into n linear factors. • In this case, the partial fractions decomposition of R(x) is…

Case 2 • Q(x) has only linear factors, including some repeated factors. • Suppose Q(x) contains the factor (xa) a total of n times • i.e., Q(x) contains the factor (xa)n • The partial fractions decomposition of R(x) must include:

Case 3 • Q(x) contains an irreducible quadratic polynomial. • ax2 + bx + c, where b2  4ac < 0. • Has no real roots. • The partial fractions decomposition of R(x) must include:

Case 4 • Q(x) contains an irreducible quadratic polynomial, raised to the nth power. • (ax2 + bx + c)n, where b2  4ac < 0. • The partial fractions decomposition of R(x) must include:

8.2: Partial Fractions