Answer: To find the partial fraction decomposition of (x+6)/(x(x+1)), we can follow these steps:
Step 1: Factorize the denominator
We start by factoring the denominator, x(x+1), into its linear factors. In this case, the factors are x and (x+1).
Step 2: Write the partial fraction decomposition
Next, we express the given rational function as a sum of fractions with denominators corresponding to the linear factors of the denominator. Since we have two linear factors, we'll have two fractions in the decomposition.
Let's denote the partial fractions as A/x and B/(x+1), where A and B are constants that we need to find.
So, the partial fraction decomposition of (x+6)/(x(x+1)) can be written as:
(x+6)/(x(x+1)) = A/x + B/(x+1)
Step 3: Find the constants A and B
To determine the values of A and B, we need to find a common denominator for the right-hand side of the equation. In this case, the common denominator is x(x+1).
Multiplying both sides of the equation by x(x+1), we get:
(x+6) = A(x+1) + Bx
Expanding the right-hand side and simplifying, we have:
x + 6 = Ax + A + Bx
Combining like terms, we get:
x + 6 = (A + B)x + A
Since the left-hand side and right-hand side must be equal for all values of x, the coefficients of the corresponding powers of x on both sides of the equation must be equal. This gives us a system of equations:
1. Coefficient of x terms: 1 = A + B
2. Coefficient of constant terms: 6 = A
From equation 2, we find that A = 6.
Substituting A = 6 into equation 1, we can solve for B:
1 = 6 + B
B = 1 - 6
B = -5
So, the values of A and B are A = 6 and B = -5, respectively.
Step 4: Finalize the partial fraction decomposition
Now that we have found the values of A and B, we can rewrite the partial fraction decomposition as:
(x+6)/(x(x+1)) = 6/x - 5/(x+1)
And that's the final partial fraction decomposition of (x+6)/(x(x+1)).