Resolve in partial fractions (x+6)/((x^2+1)(x-2)^2)
Using let x = ...
I get
(27x+14)/(x^2+1) - (27/25)/(x-2) + (8/5)/(x-2)^2
how did you work this out please ?
Review your text on partial fractions. You know that the sum will be
(Ax+B)/(x^2+1) + C/(x-2) + D/(x-2)^2
If you put all that over a common denominator, you get
(Ax+B)(x-2)^2 + C(x-2)(x^2+1) + D(x^2+1)
------------------------------------------------
(x^2+1)(x-2)^2
Expand everything in the numerator and you get
(A+C)x^3 + (-4A+B-2C+D)x^2 + (4A-4B+C)x + (4B-2C+D)
------------------------------------
(x^2+1)(x-2)^2
If that is identical to (x+6)/((x^2+1)(x-2)^2)), then we must have all the coefficients in the two numerators be identical. That is,
A+C=0
-4A+B-2C+D = 0
4A-4B+C = 1
4B-2C+D = -6
If I haven't made any typos, if you solve that system you get the numbers I presented earlier.
To resolve the given rational function into partial fractions, we need to decompose it into simpler fractions. Let's begin with the given equation:
(x + 6) / ((x^2 + 1)(x - 2)^2)
Step 1: Factorize the denominator
The denominator can be factored as follows:
(x^2 + 1) = (x^2 + 2x - 2x + 1) = (x(x + 2) - 1(x + 2)) = (x - 1)(x + 2)
(x - 2)^2 = (x - 2)(x - 2)
So, the denominator becomes:
((x - 1)(x + 2))(x - 2)^2
Step 2: Express the given fraction using partial fractions
The general form for partial fractions is:
A / (x - 1) + B / (x + 2) + C / (x - 2) + D / (x - 2)^2
Step 3: Clear the denominators
Multiply both sides of the equation by the common denominator, which is ((x - 1)(x + 2))(x - 2)^2:
(x + 6) = A(x + 2)(x - 2)^2 + B(x - 1)(x - 2)^2 + C(x - 1)(x + 2)(x - 2) + D(x - 1)(x + 2)
Step 4: Solve for the unknowns
To find the values of A, B, C, and D, we substitute appropriate values of x to eliminate other unknowns one at a time.
For A, substitute x = 1:
7 = A(1 + 2)(1 - 2)^2
7 = 3A
So, A = 7/3
For B, substitute x = -2:
4 = B(-2 - 1)(-2 - 2)^2
4 = B(-3)(16)
4 = -48B
So, B = -1/12
For C, substitute x = 2:
8 = C(1 - 1)(2 + 2)(2 - 2)
8 = 0
Hence, we cannot determine the value of C as it becomes 0.
For D, let's solve the equation for D.
Dx^2 - 3Dx + 2D = x + 6
Comparing coefficients, we get
D = 1
-3D = 1
2D = 6
From the second equation, D = -1/3
Step 5: Write the resolved equation
Now, we can write the resolved equation using the values of A, B, C (0), and D:
(x + 6) / ((x^2 + 1)(x - 2)^2) = 7/3 / (x - 1) - 1/12 / (x + 2) - 1/3 / (x - 2) + 1 / (x - 2)^2
So, the resolved equation in partial fractions is:
7/3 / (x - 1) - 1/12 / (x + 2) - 1/3 / (x - 2) + 1 / (x - 2)^2