express x-1/(x+1)(x-2)^2 into partial fractions
let
(x-1)/((x+1)(x-2)^2) = A/(x-1) + B/(x-2) + C/(x-2)^2
multiply both sides by (x+1)(x-2)^2
A(x-2)^2 + B(x-2)(x+1) + C(x+1) = x-1
let x = 2, ---> 3C = 1 or C=13
let x = -1 ---> 9A + -2 or A = -2/9
let x = 0 ----> 4A - 2B + C = -1
4(-2/9) - 2B + 1/3 = -1
times 9
-8 - 18B + 3 = -9
B = 2/9
so (x-1)/(x-2)^2 = -2/(9(x+1)) + 2/(9(x-2)) + 1/(3(x-2)^2)
verified with Wolfram:
http://www.wolframalpha.com/input/?i=%28x-1%29%2F%28%28x%2B1%29%28x-2%29%5E2%29
in the middle of the solution above the line should read:
et x = 2, ---> 3C = 1 or C=1/3
To express the expression x - 1 / (x + 1)(x - 2)^2 into partial fractions, you can follow the following steps:
Step 1: Factorize the denominator:
(x + 1)(x - 2)^2 = (x + 1)(x - 2)(x - 2)
Step 2: Express the given expression as a sum of partial fractions with unknown numerators:
x - 1 / (x + 1)(x - 2)^2 = A / (x + 1) + B / (x - 2) + C / (x - 2)^2
Step 3: Multiply through the denominator on both sides of the equation to clear the fractions:
x - 1 = A(x - 2)(x - 2) + B(x + 1)(x - 2) + C(x + 1)
Step 4: Expand and collect like terms on the right-hand side of the equation:
x - 1 = A(x^2 - 4x + 4) + B(x^2 - x - 2x + 2) + C(x + 1)
Step 5: Simplify and rearrange the equation:
x - 1 = A(x^2 - 4x + 4) + B(x^2 - 3x + 2) + C(x + 1)
x - 1 = (A + B) x^2 + (-4A - 3B + C) x + (4A + 2B + C)
Step 6: Equate the coefficients of like powers of x on both sides of the equation:
1) Coefficients of x^2: A + B = 0
2) Coefficients of x: -4A - 3B + C = 1
3) Coefficients of x^0: 4A + 2B + C = -1
Step 7: Solve the system of equations to find the values of A, B, and C:
From equation 1), we have A = -B
Substituting A = -B into equations 2) and 3):
-4(-B) - 3B + C = 1
4(-B) + 2B + C = -1
Expanding and simplifying, we have:
4B - 3B + C = 1
-4B + 2B + C = -1
B - C = 1
-2B + C = -1
Adding the two equations, we get:
-B = 0
From this, we can conclude that B = 0.
Substituting B = 0 into equation B - C = 1, we have:
0 - C = 1
Solving for C, we get: C = -1
Finally, substituting B = 0 and C = -1 into equation A = -B, we have:
A = -0
Therefore, the partial fraction decomposition of x - 1 / (x + 1)(x - 2)^2 is:
x - 1 / (x + 1)(x - 2)^2 = 0 / (x + 1) + 0 / (x - 2) - 1 / (x - 2)^2
So, the partial fraction decomposition is 0 + 0 - 1 / (x - 2)^2.