University:
York University
Course:
MATH 1013 | Applied Calculus
Academic year:

2023
Views:

79

Pages:

2
Author:

Niklas

II. Rational Functions PARTIAL FRACTION DECOMPOSITION (Ch. 7-4) I. FRACTIONS WITH DISTINCT LINEAR FACTORS 8x - 42 / x^2 + 3x - 18 8x - 42 / x^2 + 3x - 18 = 8x - 42 / (x + 6)(x - 3) 8x - 42 / x^2 + 3x - 18 = A / (x + 6) + B / (x - 3) 8x - 42 = A(x - 3) + B(x + 6) let x = 3: 8(3) - 42 = A(3 - 3) + B(3 * 6) -18 = 9B B = -2 let x = -6: 8(-6) - 42 = A(-6 - 3) * B(-6 + 6) -90 = -9A A = 10 8x - 42 / x^2 + 3x - 18 = 10 / (x + 6) - 2 / (x - 3) II. FRACTIONS WITH REPEATED LINEAR FACTORS 5x^2 + 20x + 6 / x(x + 1)^2 5x^2 + 20x + 6 / x(x + 1)^2 = A / x + B / (x + 1) + C / (x + 1)^2 5x^2 + 20x + 6 = A(x - 1)^2 + Bx(x + 1) + Cx let x = -1: 5(-1)^2 + 20(-1) + 6 = A(6) * B(-1)(6) - C -9 = -C C = 9 let x = 0: 0 + 0 * 6 = A(i) * B()() * C() A = 6 let x = 1, A = 6, C = 9^2: 5 + 20 + 6 = 6(4) + 2B + 9(i) 31 = 24 + 2B + 9 -2 = 2B B = -1 5y^2 + 20x + 6 / x(x + 1)^2 = 6 / x - 1 / (x + 1) + 9 / (x + 1)^2 Section III (3x^2 + 4x + 4) / (x^3 + 4x) (3x^2 + 4x + 4) / (x^3 + 4x) = (3x^2 + 4x + 4) / (x(x^2 + 4)) (3x^2 + 4x + 4) / (x^3 + 4x) = A/x + (Bx + C) / (x^2 + 4) 3x^2 + 4x + 4 = A(x^2 + 4) + (Bx + C)x = Ax^2 + 4A + Bx^2 + Cx = (A + B)x^2 + (C + 4A) A + B = 3 C = 4 4A = 4 B = 2 C = 4 A = 1 (3x^2 + 4x + 4) / (x^3 + 4x) = 1/x + (2x + 4) / (x^2 + 4) Section IV (8x^3 + 13x) / ((x^2 + 2)^2) (8x^3 + 13x) / ((x^2 + 2)^2) = (Ax + B) / (x^2 + 2) + (Cx + D) / ((x^2 + 2)^2) 8x^3 + 13x = (Ax + B)(x^2 + 2) + (Cx + D) = Ax^3 + 2Ax + Bx^2 + 2B + Cx + D = Ax^3 + Bx^2 + (2A + C)x + (2B + D) A = 8 B = 0 2A + C = 13 2B + D = 0 A = 8 B = 0 C = -3 D = 0 (8x^3 + 13x) / ((x^2 + 2)^2) = 8x / (x^2 + 2) - 3x / ((x^2 + 2)^2) Section V (x + 5) / (x^2(x^2 + 1)^2) (x + 5) / (x^2(x^2 + 1)^2) = A/x + B/x^2 + (Cx + D) / (x^2 + 1) + (Ex + F) / ((x^2 + 1)^2) x + 5 = Ax(x^2 + 1)^2 + B(x^2 + 1)^2 + (Cx + D)x^2(x^2 + 1) + (Ex + F)x^2 = Ax^5 + 2Ax^3 + Ax + Bx^4 + 2Bx^2 + B + Cx^5 + Dx^4 + Dx^2 + Ex^3 + Fx^2 A = 1 B = 5 C = 1 D = -5 E = -1 F = -5 (x + 5) / (x^2(x^2 + 1)^2) = 1/x + 5/x^2 - (x + 5) / (x^2 + 1) - (x + 5) / ((x^2 + 1)^2)