925 9.4 Partial Fractions In summary, to solve for the constants in the numerators of a partial fraction decomposition, use either of the following methods or a combination of the two. Techniques for Decomposition into Partial Fractions Method 1 For Linear Factors Step 1 Multiply each side of the resulting rational equation by the common denominator. Step 2 Substitute the zero of each factor in the resulting equation. For repeated linear factors, substitute as many other numbers as necessary to find all the constants in the numerators. The number of substitutions required will equal the number of constants A, B, c . Method 2 For Quadratic Factors Step 1 Multiply each side of the resulting rational equation by the common denominator. Step 2 Collect like terms on the right side of the equation. Step 3 Equate the coefficients of like terms to form a system of equations. Step 4 Solve the system to find the constants in the numerators. 9.4 Exercises CONCEPT PREVIEW Answer each question. 1. By what expression should we multiply each side of 5 3x12x + 12 = A 3x + B 2x + 1 so that there are no fractions in the equation? 2. In Exercise 1, after clearing fractions to decompose, the equation A12x + 12 + B13x2 = 5 results. If we let x = 0, what is the value of A? 3. By what expression should we multiply each side of 3x - 2 1x + 4213x2 + 12 = A x + 4 + Bx + C 3x2 + 1 so that there are no fractions in the equation? 4. In Exercise 3, after clearing fractions to decompose, the equation 3x - 2 = A13x2 + 12 + 1Bx + C21x + 42 results. If we let x = -4, what is the value of A? 5. By what expression should we multiply each side of 3x - 1 x12x2 + 122 = A x + Bx + C 2x2 + 1 + Dx + E 12x2 + 122 so that there are no fractions in the equation?
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