Problem 33 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{x+4}{x^{2}\left(x^{2}+4\right)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{x+4}{x^{2}(x^{2}+4)}\) is \(\frac{1}{x^{2}} + \frac{x+1}{x^{2}+4}\).

Step 1: Identify the form of the denominator

The given expression is of the form \(\frac{A}{x^{2}} + \frac{Bx+C}{x^{2}+4}\). This is because the denominator \(x^2(x^{2}+4)\) contains two separate terms, namely \(x^2\) and \(x^2 + 4\). Each of these terms contributes to a component in the decomposition.

Step 2: Write the partial fraction decomposition

Next, write down the form the decomposition must take: \(\frac{x+4}{x^{2}(x^{2}+4)} = \frac{A}{x^{2}} + \frac{Bx+C}{x^{2}+4}\).

Step 3: Clear the fractions

Subtract \(A/(x^2)\) and \((Bx+C)/(x^2+4)\) from both sides and multiply every term by \(x^2(x^2 + 4)\) to clear the fractions: \(x+4 = A(x^2 + 4) + Bx(x^2) + Cx^2\).

Step 4: Solve the resulting equation

Solving the resulting equation gives A = 1, B = 0, C = 1.

Step 5: Substituting A, B, and C into the decomposition

Substitute the values of A, B, and C to the decomposition, we get \(\frac{x+4}{x^{2}(x^{2}+4)} = \frac{1}{x^2} + \frac{x+1}{x^2+4}\).