Explanation:
Here's how you can use partial fraction decomposition to write (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) as the sum of simpler terms:
Factor the denominator to simplify the expression: x^2 + x - 2 = (x - 1)(x + 2)
Write the partial fraction decomposition of the integrand as the sum of simpler terms: (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) = A/(x - 1) + B/(x + 2) + Cx + D, where A, B, C, and D are constants.
Multiply both sides of the equation by (x^2 + x - 2) to find the values of A, B, C, and D: x^3 + 2x^2 + 3x + 4 = A(x + 2) + B(x - 1) + (Cx + D)(x^2 + x - 2)
Substitute x = 1 and x = -2 into the equation to find two equations for A, B, C, and D:
x = 1: 4 + 2 + 3 + 4 = 9 = A(-2) + B + (C + D)(-1)
x = -2: -8 - 4 + 6 - 8 = -16 = A(1) + B(-2) + (C - 2D)(4)
Solve the system of equations to find the values of A, B, C, and D:
A = 9/3, B = 5, C = -11/3, D = 7/3
So the partial fraction decomposition of (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) is (9/3)/(x - 1) + (5)/(x + 2) - (11/3)x + (7/3).