920 CHAPTER 9 Systems and Matrices 9.4 Partial Fractions ■ Decomposition of Rational Expressions ■ Distinct Linear Factors ■ Repeated Linear Factors ■ Distinct Linear and Quadratic Factors ■ Repeated Quadratic Factors Decomposition of Rational Expressions The sums of rational expressions are found by combining two or more rational expressions into one rational expression. Here, the reverse process is considered: Given one rational expression, express it as the sum of two or more rational expressions. A special type of sum involving rational expressions is a partial fraction decomposition—each term in the sum is a partial fraction. Add rational expressions 2 x + 1 + 3 x = 5x + 3 x1x + 12 Partial fraction decomposition To find the constants in Step 5, the goal is to form a system of equations with as many equations as there are unknowns in the numerators. Partial Fraction Decomposition of f 1 x2 g 1 x2 To form a partial fraction decomposition of a rational expression, follow these steps. Step 1 If ƒ1x2 g1x2 is not a proper fraction (a fraction with the numerator of lesser degree than the denominator), divide ƒ1x2 by g1x2. For example, x4 - 3x3 + x2 + 5x x2 + 3 = x2 - 3x - 2 + 14x + 6 x2 + 3 . Then apply the following steps to the remainder, which is a proper fraction. Step 2 Factor the denominator g1x2 completely into factors of the form 1ax + b2m or 1cx2 + dx + e2n, where cx2 + dx + e is not factorable and m and n are positive integers. Step 3 (a) For each distinct linear factor 1ax + b2, the decomposition must include the term A ax + b . (b) For each repeated linear factor 1ax + b2m, the decomposition must include the terms A1 ax + b + A2 1ax + b22 + g+ Am 1ax + b2m . Step 4 (a) For each distinct quadratic factor 1cx2 + dx + e2, the decomposition must include the term Bx + C cx2 + dx + e . (b) For each repeated quadratic factor 1cx2 + dx + e2n, the decomposition must include the terms B1 x + C1 cx2 + dx + e + B2 x + C2 1cx2 + dx + e22 + g+ Bn x + Cn 1cx2 + dx + e2n . Step 5 Use algebraic techniques to solve for the constants in the numerators of the decomposition. LOOKING AHEAD TO CALCULUS In calculus, partial fraction decomposition provides a powerful technique for determining integrals of rational functions.
RkJQdWJsaXNoZXIy NjM5ODQ=