980 CHAPTER 9 Systems and Matrices Concepts Examples 9.4 Partial Fractions 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. Method 1 For Linear Factors Step 1 Multiply each side 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 by the common denominator. Step 2 Collect like terms on the right side of the resulting 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. Find the partial fraction decomposition of 9 2x2 + 9x + 9 . 9 12x + 321x + 32 = A 2x + 3 + B x + 3 (1) Multiply by 12x + 321x + 32. 9 = A1x + 32 + B12x + 32 9 = Ax + 3A + 2Bx + 3B 9 = 1A + 2B2x + 13A + 3B2 Now solve the system A + 2B = 0 3A + 3B = 9 to obtain A = 6 and B = -3. 9 2x2 + 9x + 9 = 6 2x + 3 + -3 x + 3 Substitute into (1). Check this result by combining the terms on the right. 9.5 Nonlinear Systems of Equations Solving a Nonlinear System of Equations A nonlinear system can be solved by the substitution method, the elimination method, or a combination of the two methods. Visualizing the types of graphs involved in a nonlinear system helps predict the possible numbers of ordered pairs of real numbers that may be in the solution set of the system. x y 0 This system has four real solutions because there are four points of intersection. Solve the system. x2 + 2xy - y2 = 14 (1) x2 - y2 = -16 (2) x2 + 2xy - y2 = 14 (1) -x2 + y2 = 16 Multiply (2) by -1. 2xy = 30 Add to eliminate x2 and y2. xy = 15 Divide by 2. y = 15 x Solve for y. Substitute 15 x for y in equation (2). x2 - a 15 x b 2 = -16 (2) with y = 15 x x2 - 225 x2 = -16 Square. x4 + 16x2 - 225 = 0 Multiply by x2. Add 16x2. 1x2 - 921x2 + 252 = 0 Factor. x2 - 9 = 0 or x2 + 25 = 0 Zero-factor property x = {3 or x = {5i Solve each equation. Find corresponding y-values. The solution set is 513, 52, 1-3, -52, 15i, -3i2, 1-5i, 3i26.
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