Resolving 3x2/(x2x3-x) into Partial Fractions
Partial fraction decomposition is a technique in algebra that allows us to break down a complex fraction into a sum of simpler fractions. This method is particularly useful in integration, simplification, and solving differential equations. In this article, we will walk through an example of resolving 3x2/(x2x3-x) into partial fractions.
Step-by-Step Guide to Decomposition
Let's begin by expressing the fraction 3x2/(x2x3-x) in the required form:
frac3x2/x2x3-x A/x Bx C/x2x3
Or 3x2 A(x2) Bx(x3) C(x2x3-x)
Choosing the Values of x for Equations
1. First, we will find A. To do this, we multiply both sides by the denominator x2x3-x and set x -1:
3(-1)2 A(-12) B(-1)(-13) C(-12(-1)3 - (-1))
3 -1A - 1B 2C
From this, we get:
-1A - B 2C 3 (i)
2. Next, we find C. We substitute x 0:
3(0)2 A(02) B(0)(03) C(02(03-0))
0 C
From this, we get:
C 3
3. Now, we find B. We substitute x 1:
3(1)2 A(12) B(1)(13) C(12(13-1))
3 A B 2C
From this, we get:
A B 2C 3 (ii)
Solving the System of Equations
From (i) and (ii), we have:
-1A - B 2C 3 (i)
A B 2C 3 (ii)
Subtract (i) from (ii):
2A 2B 0
A B 0 rArr; B -A
Substitute B -1/3 (from earlier steps) and solve for A:
-1/3 B 2C 3
-1/3 -1/3 2*3 3
Therefore, the partial fraction decomposition is:
3x2/(x2x3-x) -1/3/x 1/3x(3/x2x3)
Conclusion
Through the process of partial fraction decomposition, we have successfully simplified the fraction 3x2/(x2x3-x) into its constituent parts, making it easier to handle in further calculations.