Express 13-15x^2/(x^2-1)(x 3) into partial fractions
Assuming you meant
(13-15x^2)/((x^2-1)(x+3))
then you get
(1/2)/(x+1) - (1/4)/(x-1) - (61/4)/(x+3)
If you mean x-3, then just make the fix. I trust you just wanted confirmation of your work, eh? If not, and you got stuck, how far did you get?
To express the given expression as partial fractions, we first need the denominator of the expression factored completely:
The denominator is (x^2-1)(x+3).
The first step is to factor the trinomial x^2-1. This is a difference of squares, so it can be factored as (x-1)(x+1).
Now the denominator is ((x-1)(x+1))(x+3).
Since this is a quadratic denominator, we will have two partial fractions.
Let's assume that the expression can be written as:
13-15x^2/((x-1)(x+1)(x+3)) = A/(x-1) + B/(x+1) + C/(x+3)
Now, we need to find the values of A, B, and C.
To do this, we proceed as follows:
Multiply both sides of the equation by (x-1)(x+1)(x+3):
13-15x^2 = A(x+1)(x+3) + B(x-1)(x+3) + C(x-1)(x+1)
Expanding the right side:
13-15x^2 = A(x^2 + 4x + 3) + B(x^2 + 2x - 3) + C(x^2 - 1)
Now, we can group the terms with the same powers of x:
13-15x^2 = (A+B+C)x^2 + (4A+2B)x + (3A-3B-C)
Equating the coefficients of each power of x, we have the following equations:
1. Coefficient of x^2: -15 = A + B + C
2. Coefficient of x: 0 = 4A + 2B
3. Constant term: 13 = 3A - 3B - C
From equation 2, we can solve for B:
0 = 4A + 2B
-2B = 4A
B = -2A
Substituting B = -2A into equation 1:
-15 = A - 2A + C
-15 = -A + C
C = -15 + A
Substituting the values of B and C into equation 3:
13 = 3A - 3(-2A) - (-15 + A)
13 = 3A + 6A + 15 - A
13 = 8A + 15
-2 = 8A
A = -1/4
Substituting A = -1/4 into B and C:
B = -2A
B = -2(-1/4)
B = 1/2
C = -15 + A
C = -15 + (-1/4)
C = -61/4
Therefore, the expression can be written as partial fractions as:
13-15x^2/((x-1)(x+1)(x+3)) = -1/4(x-1) + 1/2(x+1) - 61/4(x+3)
To summarize the steps:
1. Factor the denominator completely.
2. Assume the expression can be written as a sum of partial fractions.
3. Multiply both sides by the denominator.
4. Expand and group terms with the same powers of x.
5. Equate coefficients to get a system of equations.
6. Solve the system of equations to find the values of the unknown coefficients.