How do I solve this via partial fraction decomposition?
(x^2+9)/(x^4-2x^2-8)
The bottom factors to (x^2-4)(x^2+2)
= (x+2)x-2)x^2+4)
so let
(x^2+9)/(x^4-2x^2-8) = A/(x+2) + B/(x-2) + C/(x^2+2)
( A(x-2)(x^2+2) + B(x+2)(x^2+2) + C(x+2)(x-2) ) = x^2 + 9
since the denominators on the left and right are the same.
This must be true for all values of x
let x=2 ---> 24B = 13 or B = 13/24
let x=-2 --> - 24A = 13 or A = -13/24
let x = 1 --> -3A + 9B - 3C = 10
sub in the values of A and B to get
C = -7/6
so (x^2+9)/(x^4-2x^2-8)
= -13/(24(x+2)) + 13/(24(x-2)) - 7/(6(x^2+2))
Thank you. But what about the "Cx+D" part?
You are correct, Micki, that there is supposed to be a (Cx+D) over a quadratic denominator. Luckily, in this case, C=0, so it worked out. Had it been otherwise, I'm sure it would have become apparent.
To solve this expression using partial fraction decomposition, we need to decompose the given rational function into simpler fractions. Here are the steps you can follow:
Step 1: Factorize the denominator.
To factorize the denominator x^4-2x^2-8, we can treat it as a quadratic equation in terms of x^2. So let's substitute y = x^2 and rewrite the equation as y^2 - 2y - 8 = 0. Now we can factorize this quadratic equation as (y - 4)(y + 2) = 0. Substituting back x^2 for y, we get (x^2 - 4)(x^2 + 2) = 0.
Step 2: Set up the partial fraction decomposition.
Now that we have factorized the denominator, we can express the given rational function as a sum of simpler fractions:
(x^2+9)/(x^4-2x^2-8) = A/(x^2 - 4) + B/(x^2 + 2)
Step 3: Solve for the constants A and B.
To solve for the constants A and B, we need to find the values that make the equation true for all x. To do this, we need to find a common denominator and then equate the numerators:
(x^2+9) = A(x^2 + 2) + B(x^2 - 4)
Expanding the right side:
x^2 + 9 = (A + B)x^2 + (2A - 4B)
By comparing the coefficients of like terms, we can set up a system of equations:
1. For the x^2 terms: A + B = 1
2. For the constant terms: 2A - 4B = 9
Step 4: Solve the system of equations.
Solve the system of equations to find the values of A and B.
In this case, the first equation can be rearranged as B = 1 - A. Substituting this into the second equation:
2A - 4(1 - A) = 9
2A - 4 + 4A = 9
6A - 4 = 9
6A = 13
A = 13/6
Substituting the value of A into the first equation:
13/6 + B = 1
B = 1 - 13/6
B = 6/6 - 13/6
B = -7/6
So the constants A = 13/6 and B = -7/6.
Step 5: Write the partial fraction decomposition.
Now that we have found the constants A and B, we can write the partial fraction decomposition of the rational function:
(x^2+9)/(x^4-2x^2-8) = (13/6)/(x^2 - 4) - (7/6)/(x^2 + 2)
And that's it! You have successfully solved the given expression using partial fraction decomposition.