This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's the solution for Question Q1: Q1 a) We need to resolve (x - 11)/(x^2 - x - 2) into partial fractions. Step 1: Factor the denominator. x^2 - x - 2 = (x-2)(x+1) Step 2: Set up the partial fraction decomposition. Since the denominator has distinct linear factors, we write: (x - 11)/((x-2)(x+1)) = (A)/(x-2) + (B)/(x+1) Step 3: Combine the fractions on the right side. (x - 11)/((x-2)(x+1)) = (A(x+1) + B(x-2))/((x-2)(x+1)) Step 4: Equate the numerators. x - 11 = A(x+1) + B(x-2) Step 5: Solve for A and B by substituting values for x. Let x = 2: 2 - 11 = A(2+1) + B(2-2) -9 = 3A + 0 A = -3 Let x = -1: -1 - 11 = A(-1+1) + B(-1-2) -12 = 0 - 3B -12 = -3B B = 4 Step 6: Write the partial fraction decomposition. (x - 11)/(x^2 - x - 2) = (-3)/(x-2) + (4)/(x+1) The partial fraction decomposition is (4)/(x+1) - (3)/(x-2) Q1 b) We need to resolve (3 - x)/((x^2 + 3)(x + 3)) into partial fractions. Step 1: Identify the types of factors in the denominator. The denominator has an irreducible quadratic factor (x^2 + 3) and a linear factor (x+3). Step 2: Set up the partial fraction decomposition. (3 - x)/((x^2 + 3)(x + 3)) = (Ax + B)/(x^2 + 3) + (C)/(x+3) Step 3: Combine the fractions on the right side. (3 - x)/((x^2 + 3)(x + 3)) = ((Ax + B)(x+3) + C(x^2 + 3))/((x^2 + 3)(x + 3)) Step 4: Equate the numerators. 3 - x = (Ax + B)(x+3) + C(x^2 + 3) Step 5: Solve for A, B, and C. Let x = -3: 3 - (-3) = (A(-3) + B)(-3+3) + C((-3)^2 + 3) 6 = 0 + C(9+3) 6 = 12C C = (6)/(12) = (1)/(2) Expand the equation and group terms by powers of x: 3 - x = Ax^2 + 3Ax + Bx + 3B + Cx^2 + 3C 3 - x = (A+C)x^2 + (3A+B)x + (3B+3C) Compare coefficients: Coefficient of x^2: 0 = A+C Since C = (1)/(2), then A = -(1)/(2). Coefficient of x: -1 = 3A+B Substitute A = -(1)/(2): -1 = 3(-(1)/(2)) + B -1 = -(3)/(2) + B B = -1 + (3)/(2) = (1)/(2) (Check constant term: 3 = 3B+3C = 3((1)/(2)) + 3((1)/(2)) = (3)/(2) + (3)/(2) = (6)/(2) = 3. This is consistent.) Step 6: Write the partial fraction decomposition. (3 - x)/((x^2 + 3)(x + 3)) = (-1)/(2)x + (1)/(2)x^2 + 3 + (1)/(2)x+3 (3 - x)/((x^2 + 3)(x + 3)) = (1-x)/(2(x^2 + 3)) + (1)/(2(x+3)) The partial fraction decomposition is (1-x)/(2(x^2 + 3)) + (1)/(2(x+3)) Q1 c) We need to resolve (x^3 - 6x + 9)/(x^2 + x - 2) into partial fractions. Step 1: Perform polynomial long division because the degree of the numerator (3) is greater than the degree of the denominator (2). c|cc cc 2rx & -1 \\ 2-5 x^2+x-2 & x^3 & +0x^2 & -6x & +9 \\ 2r-(x^3 & +x^2 & -2x) \\ 2-4 2r0 & -x^2 & -4x & +9 \\ 2r & -(-x^2 & -x & +2) \\ 3-5 2r & 0 & -3x & +7 \\ So, (x^3 - 6x + 9)/(x^2 + x - 2) = x - 1 + (-3x + 7)/(x^2 + x - 2). Step 2: Factor the denominator of the remainder term. x^2 + x - 2 = (x+2)(x-1) Step 3: Set up the partial fraction decomposition for the remainder term. (-3x + 7)/((x+2)(x-1)) = (A)/(x+2) + (B)/(x-1) Step 4: Combine the fractions on the right side. (-3x + 7)/((x+2)(x-1)) = (A(x-1) + B(x+2))/((x+2)(x-1)) Step 5: Equate the numerators. -3x + 7 = A(x-1) + B(x+2) Step 6: Solve for A and B by substituting values for x. Let x = 1: -3(1) + 7 = A(1-1) + B(1+2) 4 = 0 + 3B B = (4)/(3) Let x = -2: -3(-2) + 7 = A(-2-1) + B(-2+2) 6 + 7 = -3A + 0 13 = -3A A = -(13)/(3) Step 7: Write the complete partial fraction decomposition. (x^3 - 6x + 9)/(x^2 + x - 2) = x - 1 + (-13)/(3)x+2 + (4)/(3)x-1 (x^3 - 6x + 9)/(x^2 + x - 2) = x - 1 - (13)/(3(x+2)) + (4)/(3(x-1)) The partial fraction decomposition is x - 1 - (13)/(3(x+2)) + (4)/(3(x-1)) Got more? Send 'em.