Express (x^2+23)/((x+1)^3(x-2)) in partial fraction. Thanks very much
To express the given rational function as a sum of partial fractions, follow these steps:
Step 1: Factorize the denominator.
In this case, the denominator is (x+1)^3(x-2).
Step 2: Write the partial fraction decomposition.
The partial fraction decomposition of the rational function is given as:
(x^2+23)/((x+1)^3(x-2)) = A/(x+1) + B/(x+1)^2 + C/(x+1)^3 + D/(x-2)
Here, A, B, C, and D are constants that need to be determined.
Step 3: Clear the fractions.
Multiply both sides of the equation by the denominator (x+1)^3(x-2) to eliminate the fractions. You will then have:
(x^2+23) = A(x+1)^2(x-2) + B(x+1)(x-2) + C(x-2) + D(x+1)^3
Step 4: Solve for constants.
The next step is to solve for the constants A, B, C, and D.
To do this, expand and simplify the right-hand side of the equation to gather like terms, then equate the coefficients of the corresponding powers of x.
Expand and simplify:
(A(x+1)^2(x-2) + B(x+1)(x-2) + C(x-2) + D(x+1)^3)
= (A(x^3 - 2x^2 + 3x - 2) + B(x^2 - x - 2) + C(x-2) + D(x^3 + 3x^2 + 3x + 1))
= (Ax^3 + (3A + B + C)x^2 + (3A - B + 3C - 2C + D + 3A)x + (-2A - 2B - 2D - 2))
Equate the coefficients:
Coefficient of x^3: 0 = A + D
Coefficient of x^2: 1 = 3A + B + C
Coefficient of x^1: 0 = 3A - B + C
Coefficient of x^0: 23 = -2A - 2B - 2D - 2
Solve this system of equations to find the values of A, B, C, and D.
Step 5: Substitute the found constants back into the partial fraction decomposition.
Once you have determined the values of A, B, C, and D, substitute these values back into the partial fraction decomposition equation:
(x^2+23)/((x+1)^3(x-2)) = A/(x+1) + B/(x+1)^2 + C/(x+1)^3 + D/(x-2)
Now you have the expression in partial fraction form.