Chapter 9 - Systems and Matrices - 9.4 Partial Fractions - 9.4 Exercises - Page 895: 33

$$ - \frac{1}{{x + 2}} + \frac{3}{{{{\left( {{x^2} + 4} \right)}^2}}}$$

$$\eqalign{ & \frac{{ - {x^4} - 8{x^2} + 3x - 10}}{{\left( {x + 2} \right){{\left( {{x^2} + 4} \right)}^2}}} \cr & {\text{The partial fraction decomposition is }} \cr & \frac{{ - {x^4} - 8{x^2} + 3x - 10}}{{\left( {x + 2} \right){{\left( {{x^2} + 4} \right)}^2}}} = \frac{A}{{x + 2}} + \frac{{Bx + C}}{{{x^2} + 4}} + \frac{{Dx + E}}{{{{\left( {{x^2} + 4} \right)}^2}}} \cr & {\text{Multiply each side by }}\left( {x + 2} \right){\left( {{x^2} + 4} \right)^2} \cr & - {x^4} - 8{x^2} + 3x - 10 = A{\left( {{x^2} + 4} \right)^2} + \left( {x + 2} \right)\left( {{x^2} + 4} \right)\left( {Bx + C} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {x + 2} \right)\left( {Dx + E} \right)\,\,\,\,\,\,\left( 1 \right) \cr & {\text{Expand and combine like terms on the right of }}\left( 1 \right) \cr & - {x^4} - 8{x^2} + 3x - 10 = A\left( {{x^4} + 8{x^2} + 16} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {{x^3} + 4x + 2{x^2} + 8} \right)\left( {Bx + C} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + D{x^2} + Ex + 2Dx + 2E \cr & - {x^4} - 8{x^2} + 3x - 10 = A{x^4} + 8A{x^2} + 16A + B{x^4} + 4B{x^2} + 2B{x^3} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 8Bx + C{x^3} + 4Cx + 2C{x^2} + 8C \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + D{x^2} + Ex + 2Dx + 2E \cr & - {x^4} - 8{x^2} + 3x - 10 = \left( {A + B} \right){x^4} + \left( {2B + C} \right){x^3} + \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {8A + 4B + 2C + D} \right){x^2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {8B + 4C + E + 2D} \right)x + 8C + 16A + 2E \cr & {\text{Equating the coefficients}} \cr & A + B = - 1 \cr & 2B + C = 0 \cr & 8A + 4B + 2C + D = - 8 \cr & 8B + 4C + 2D + E = 3 \cr & 8C + 16A + 2E = - 10 \cr & {\text{Solving the system of equations using a CAS we obtain}} \cr & A = - 1,\,\,\,B = 0,\,\,\,C = 0,\,\,\,\,D = 0,\,\,\,E = 3 \cr & {\text{Use }}A,{\text{ }}B,\,C,D\,\,{\text{ and }}E{\text{ to find the partial fraction decomposition}} \cr & \frac{{ - {x^4} - 8{x^2} + 3x - 10}}{{\left( {x + 2} \right){{\left( {{x^2} + 4} \right)}^2}}} = - \frac{1}{{x + 2}} + \frac{3}{{{{\left( {{x^2} + 4} \right)}^2}}} \cr} $$