partial decomposition
12/x(x-2)(x+3)
To partially decompose the expression 12/x(x-2)(x+3), we need to break it down into simpler fractions. The process for partial fraction decomposition involves the following steps:
1. Factorize the denominator: The denominator in our expression is x(x-2)(x+3). So, let's break it down into its factors.
x(x-2)(x+3)
2. Write the expression as a sum of fractions: Since we have a third-degree polynomial in the denominator, we need to break it down into three fractions.
12/x(x-2)(x+3) = A/x + B/(x-2) + C/(x+3)
Here, A, B, and C are constants that we need to determine.
3. Clear the fractions by finding a common denominator: The common denominator for our three fractions is (x)(x-2)(x+3). Multiply each fraction by the appropriate factor to clear the fractions.
12 = A(x-2)(x+3) + B(x)(x+3) + C(x)(x-2)
4. Simplify and solve for the constants: Expand and simplify the expression on the right side of the equation.
12 = A(x^2 + x - 6) + B(x^2 + 3x) + C(x^2 - 2x)
Now, let's group the terms with the same powers of x:
12 = (A + B + C)x^2 + (A + 3B - 2C)x - 6A
Since the left side of the equation does not have any x terms, we can equate the coefficients on the right side:
A + B + C = 0 (For x^2 term)
A + 3B - 2C = 0 (For x term)
-6A = 12 (The constant term)
Solve this system of linear equations to find the values of A, B, and C.
Using any method of solving linear equations (such as substitution or elimination), we find:
A = -2
B = 1
C = 1
5. Write the expression in partial fraction form: Now that we have determined the values of A, B, and C, we can rewrite the original expression in partial fraction form.
12/x(x-2)(x+3) = -2/x + 1/(x-2) + 1/(x+3)
And that's the partial decomposition of the expression 12/x(x-2)(x+3).