Chapter 9 - Systems and Matrices - 9.4 Partial Fractions - 9.4 Exercises - Page 895: 34

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

$$\eqalign{ & \frac{{3{x^4} + {x^3} + 5{x^2} - x + 4}}{{\left( {x - 1} \right){{\left( {{x^2} + 1} \right)}^2}}} \cr & {\text{The partial fraction decomposition is }} \cr & \frac{{3{x^4} + {x^3} + 5{x^2} - x + 4}}{{\left( {x - 1} \right){{\left( {{x^2} + 1} \right)}^2}}} = \frac{A}{{x - 1}} + \frac{{Bx + C}}{{{x^2} + 1}} + \frac{{Dx + E}}{{{{\left( {{x^2} + 1} \right)}^2}}} \cr & {\text{Multiply each side by }}\left( {x - 1} \right){\left( {{x^2} + 1} \right)^2} \cr & 3{x^4} + {x^3} + 5{x^2} - x + 4 = A{\left( {{x^2} + 1} \right)^2} + \left( {x - 1} \right)\left( {{x^2} + 1} \right)\left( {Bx + C} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {x - 1} \right)\left( {Dx + E} \right) \cr & {\text{Expand and combine like terms on the right of }}\left( 1 \right) \cr & 3{x^4} + {x^3} + 5{x^2} - x + 4 = A\left( {{x^4} + 2{x^2} + 1} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {{x^3} - {x^2} + x - 1} \right)\left( {Bx + C} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + D{x^2} - Dx + Ex - E \cr & 3{x^4} + {x^3} + 5{x^2} - x + 4 = A{x^4} + 2A{x^2} + A + B{x^4} - B{x^3} + B{x^2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - Bx + C{x^3} - C{x^2} + Cx - C \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + D{x^2} - Dx + Ex - E \cr & 3{x^4} + {x^3} + 5{x^2} - x + 4 = \left( {A + B} \right){x^4} + \left( { - B + C} \right){x^3} + \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {2A + B - C + D} \right){x^2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - B + C - D + E} \right)x + \left( {A - C - E} \right) \cr & {\text{Equating the coefficients}} \cr & A + B = 3 \cr & - B + C = 1 \cr & 2A + B - C + D = 5 \cr & - B + C - D + E = - 1 \cr & A - C - E = 4 \cr & {\text{Solving the system of equations using a CAS we obtain}} \cr & A = 3,\,\,\,B = 0,\,\,\,C = 1,\,\,\,\,D = 0,\,\,\,E = - 2 \cr & {\text{Use }}A,{\text{ }}B,\,C,D\,\,{\text{ and }}E{\text{ to find the partial fraction decomposition}} \cr & \frac{{3{x^4} + {x^3} + 5{x^2} - x + 4}}{{\left( {x - 1} \right){{\left( {{x^2} + 1} \right)}^2}}} = \frac{3}{{x - 1}} + \frac{{0x + 1}}{{{x^2} + 1}} + \frac{{0x - 2}}{{{{\left( {{x^2} + 1} \right)}^2}}} \cr & \frac{{3{x^4} + {x^3} + 5{x^2} - x + 4}}{{\left( {x - 1} \right){{\left( {{x^2} + 1} \right)}^2}}} = \frac{3}{{x - 1}} + \frac{1}{{{x^2} + 1}} - \frac{2}{{{{\left( {{x^2} + 1} \right)}^2}}} \cr} $$