8.2: Partial Fractions Rational function Improper rational function

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Presentation transcript:

8.2: Partial Fractions Rational function Improper 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

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: