Resolve into partial fractions:
(x+3)(x+1)/
x(x^2+x+1)
Can anyone please help me out?
8/x^3
(x+3)(x+1)/[x(x^2+x+1)] =
A/x + (B + C x)/(x^2 + x + 1)
To find A multiply both sides by x and take the limit x --> 0:
A = 3
To find B and C you can just move the A/x term to the other side and simplify:
(B + C x)/(x^2 + x + 1) =
(x+3)(x+1)/[x(x^2+x+1)] - 3/x =
(x^2 + 4x + 3 - 3 x^2 - 3 x - 3)/[x(x^2+x+1)] =
(-2x^2 + x)/[x(x^2+x+1)] =
(1 - 2x)/(x^2+x+1)
To resolve the rational function into partial fractions, follow these steps:
Step 1: Factorize the denominator.
First, factorize the denominator, x(x^2 + x + 1), as much as possible. In this case, the denominator cannot be factored any further.
Step 2: Write the partial fraction decomposition.
Assume that the rational function can be written as:
(x + 3)(x + 1) / (x)(x^2 + x + 1) = A/x + (Bx + C) / (x^2 + x + 1)
Step 3: Find the values of A, B, and C.
Multiply both sides of the equation by the denominator, (x)(x^2 + x + 1), to eliminate the fractions:
(x + 3)(x + 1) = A(x^2 + x + 1) + (Bx + C)(x)
Step 4: Equate coefficients.
Match equivalent degree terms on both sides of the equation. This will allow you to find the values of A, B, and C.
On the left side of the equation, distribute and combine like terms:
x^2 + 4x + 3 = Ax^2 + Ax + A + Bx^2 + Cx
Combine similar degree terms:
(x^2 + Bx^2) + (Ax + Cx) + (A) = x^2(A + B) + x(A + C) + (A) = x^2 + 4x + 3
Step 5: Equate coefficients part two.
Since corresponding coefficients on each side of the equation must be equal, you can equate the coefficients of x^2, x, and the constant term:
x^2: A + B = 1
x: A + C = 4
Constant term: A = 3
Step 6: Solve the system of equations.
Solve the system of equations from Step 5. Using the first and third equations, we have A = 3 and A + B = 1. Subtracting these two equations gives B = -2. Then, substituting A = 3 into the second equation gives 3 + C = 4, so C = 1.
Step 7: Write the partial fraction solution.
Now that you have determined the values of A, B, and C, you can rewrite the original rational function with the partial fraction decomposition:
(x + 3)(x + 1) / (x)(x^2 + x + 1) = 3/x - (2x + 1) / (x^2 + x + 1)
Therefore, the resolved partial fraction form of the given rational function is 3/x - (2x + 1) / (x^2 + x + 1).