Write the partial fraction decomposition of the rational expression.
6x^2+1/x^2(x-1)^2
Because of the duplication of factors we could have
A/x + B/x^2 + C/(x-1) + D/(x-1)^2 = (6x^2 + 1)/(x^2(x-1)^2)
multiply by x^2(x-1)^1
A(x-1)^2 + Bx(x-1)^2 + Cx^2(x-1) + Dx^2 = 6x^2 + 1
let x=0 --> A = 1
let x=1 --> D = 7
then (x-1)^2 + Bx(x-1)^2 + Cx^2(x-1) + 7x^2 = 6x^2 + 1
let x=2 --> 1 + 2B + 4C + 28 = 24+1
or B + 2C = -2 (#1)
let x = -1 --> 4 - 4B - 2C + 7 = 7
or 2B + C = 2 (#2)
2 (#1) - #2
3C = -6
C = -2
then B = 2 form #1
so (6x^2 + 1)/(x^2(x-1)^2)
= 1/x^2 + 2/x - 2/(x-1) + 7/(x-1)^2
Looks like it should have said
A/x^2 + B/x + C/(x-1) + D/(x-1)^2 = (6x^2 + 1)/(x^2(x-1)^2)
but the solution from there on is correct.
To find the partial fraction decomposition of the rational expression (6x^2 + 1) / [x^2(x - 1)^2], follow these steps:
Step 1: Factor the denominator
The denominator of the rational expression is x^2(x - 1)^2. We can break it down further as follows:
x^2(x - 1)^2 = x^2(x - 1)(x - 1)
Step 2: Write the partial fraction form
The partial fraction form of the given expression is:
(6x^2 + 1) / [x^2(x - 1)^2] = A / x + B / x^2 + C / (x - 1) + D / (x - 1)^2
Note that A, B, C, and D are constants that we need to find.
Step 3: Clear the denominators
To clear the denominators and solve for A, B, C, and D, we multiply both sides of the equation by x^2(x - 1)^2:
(6x^2 + 1) = A(x - 1)^2 + B(x^2)(x - 1) + C(x^2) + D(x)(x - 1)^2
Step 4: Solve for the unknown constants
Now, we need to find the values of A, B, C, and D. We can do this by comparing the coefficients of like terms on both sides of the equation. Let's compare the coefficients of each term:
Coefficient of x^3 on the left = 0
Coefficient of x^3 on the right = B
Since there is no term with x^3 on the left side, the coefficient of x^3 must be zero on the right side. Therefore, B = 0.
Coefficient of x^2 on the left = 6
Coefficient of x^2 on the right = A + C
Comparing the coefficients, we have A + C = 6.
Coefficient of x on the left = 0
Coefficient of x on the right = -A - B + D
Comparing the coefficients, we have -A - B + D = 0.
Coefficient of the constant term (x^0) on the left = 1
Coefficient of the constant term (x^0) on the right = (A - D)
Comparing the coefficients, we have A - D = 1.
Solving this system of equations will give us the values of A, C, and D.
Step 5: Solve the system of equations
From step 4, we have the following system of equations:
A + C = 6
-A - B + D = 0
A - D = 1
Solving this system of equations will give us the values of A, C, and D.
Step 6: Substitute the values to get the final result
Once you have found the values of A, C, and D, substitute them back into the partial fraction form from step 2:
(6x^2 + 1) / [x^2(x - 1)^2] = A / x + B / x^2 + C / (x - 1) + D / (x - 1)^2
Substitute the values of A, C, and D obtained in the previous step into this equation to get the final result.