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Transcript
Partial fraction decomposition algorithm Remarkable fact: Any proper rational function (i.e. degree of numerator < degree of denominator) whose coefficients are real numbers can be written as the sum of “simple” rational functions of the form A , (ax + b)k Bx + C (ãx2 + b̃x + c̃)k̃ , where k, k̃ are non-negative integers and A, B, C, a, b, ã, b̃, c̃ are real numbers. 1. Let f (x) = p(x) q(x) be a rational function. If f is not proper (i.e. if degree p(x) ≥ degree q(x)), use long division to obtain N (x) f (x) = f˜(x) + , D(x) N (x) where f˜(x) is a polynomial and is a proper rational function. D(x) 2. Factor D into a product of linear (degree 1) and irreducible quadratic (degree 2) factors with real coefficients. Note: This is always possible in theory by the Fundamental Theorem of Algebra. 3. For each factor of D of the form (ax + b)n , expect a decomposition of the form A1 A2 An + + ··· + . ax + b (ax + b)2 (ax + b)n 4. For each factor of the form (ax2 + bx + c)m , expect a decomposition of the form B1 x + C1 B2 x + C2 Bm x + Cm + + ··· + . ax2 + bx + c (ax2 + bx + c)2 (ax2 + bx + c)m N (x) equal to the sum of the terms gotten in steps 3 and 4. D(x) 6. Multiply both sides of the equation in step 5 by D(x) and solve for the unknown constants, either by substituting in “convenient” values of the variable x or by equating the coefficients of terms with the same degree. 5. Set 1