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.
ScanSolve AI Answer
Here are the solutions to the problems. Question 8: Split (x^2)/((x^2+9)(x-3)) into partial fractions. Step 1: Set up the partial fraction decomposition. The denominator has an irreducible quadratic factor (x^2+9) and a linear factor (x-3). (x^2)/((x^2+9)(x-3)) = (Ax+B)/(x^2+9) + (C)/(x-3) Step 2: Multiply both sides by the common denominator (x^2+9)(x-3). x^2 = (Ax+B)(x-3) + C(x^2+9) Step 3: Solve for A, B, and C. To find C, let x=3: (3)^2 = (A(3)+B)(3-3) + C((3)^2+9) 9 = 0 + C(9+9) 9 = 18C C = (9)/(18) = (1)/(2) Expand the right side of the equation from Step 2: x^2 = Ax^2 - 3Ax + Bx - 3B + Cx^2 + 9C Group terms by powers of x: x^2 = (A+C)x^2 + (-3A+B)x + (-3B+9C) Equate coefficients: 1. Coefficient of x^2: A+C = 1 2. Coefficient of x: -3A+B = 0 3. Constant term: -3B+9C = 0 Substitute C=(1)/(2) into equation (1): A + (1)/(2) = 1 A = 1 - (1)/(2) = (1)/(2) Substitute A=(1)/(2) into equation (2): -3((1)/(2)) + B = 0 -(3)/(2) + B = 0 B = (3)/(2) (As a check, substitute B and C into equation (3): -3((3)/(2)) + 9((1)/(2)) = -(9)/(2) + (9)/(2) = 0. This is correct.) Step 4: Substitute the values of A, B, and C back into the partial fraction form. (x^2)/((x^2+9)(x-3)) = (1)/(2)x+(3)/(2)x^2+9 + (1)/(2)x-3 = (x+3)/(2(x^2+9)) + (1)/(2(x-3)) The partial fraction decomposition is (x+3)/(2(x^2+9)) + (1)/(2(x-3)). Question 9: Split (1)/((x-1)(x^2+4)) into partial fractions. Step 1: Set up the partial fraction decomposition. The denominator has a linear factor (x-1) and an irreducible quadratic factor (x^2+4). (1)/((x-1)(x^2+4)) = (A)/(x-1) + (Bx+C)/(x^2+4) Step 2: Multiply both sides by the common denominator (x-1)(x^2+4). 1 = A(x^2+4) + (Bx+C)(x-1) Step 3: Solve for A, B, and C. To find A, let x=1: 1 = A((1)^2+4) + (B(1)+C)(1-1) 1 = A(1+4) + 0 1 = 5A A = (1)/(5) Expand the right side of the equation from Step 2: 1 = Ax^2 + 4A + Bx^2 - Bx + Cx - C Group terms by powers of x: 1 = (A+B)x^2 + (-B+C)x + (4A-C) Equate coefficients: 1. Coefficient of x^2: A+B = 0 2. Coefficient of x: -B+C = 0 3. Constant term: 4A-C = 1 Substitute A=(1)/(5) into equation (1): (1)/(5) + B = 0 B = -(1)/(5) Substitute B=-(1)/(5) into equation (2): -(-(1)/(5)) + C = 0 (1)/(5) + C = 0 C = -(1)/(5) (As a check, substitute A and C into equation (3): 4((1)/(5)) - (-(1)/(5)) = (4)/(5) + (1)/(5) = (5)/(5) = 1. This is correct.) Step 4: Substitute the values of A, B, and C back into the partial fraction form. (1)/((x-1)(x^2+4)) = (1)/(5)x-1 + (-1)/(5)x-(1)/(5)x^2+4 = (1)/(5(x-1)) - (x+1)/(5(x^2+4)) The partial fraction decomposition is (1)/(5(x-1)) - (x+1)/(5(x^2+4)). Question 10: Evaluate (B-A) if (6x+7)/((x+3)(x+1)) = (A)/(x+3) + (B)/(x+1). Step 1: Multiply both sides by the common denominator (x+3)(x+1). 6x+7 = A(x+1) + B(x+3) Step 2: Solve for A and B. To find A, let x=-3: 6(-3)+7 = A(-3+1) + B(-3+3) -18+7 = A(-2) + 0 -11 = -2A A = (-11)/(-2) = (11)/(2) To find B, let x=-1: 6(-1)+7 = A(-1+1) + B(-1+3) -6+7 = 0 + B(2) 1 = 2B B = (1)/(2) Step 3: Evaluate (B-A). B-A = (1)/(2) - (11)/(2) B-A = (1-11)/(2) B-A = (-10)/(2) B-A = -5 The value of (B-A) is -5.