Problem 57 - Algebra and Trigonometry

Question

Explain how to find the partial fraction decomposition of a rational expression with a repeated, prime quadratic factor in the denominator.

Step-by-Step Solution

Answer

The method to find the partial fraction decomposition of a rational expression with a repeated, prime quadratic factor in the denominator is through a process starting with identifying the quadratic factor, setting up simpler fractions, matching terms with the original fraction, determining the coefficients and finally writing out the decomposition.

Step 1: Identify Repeated Quadratic Factor

Look at the denominator of the given rational expression. Identify the quadratic factor that is repeated in the denominator. Let's denote this factor as \(q(x)\).

Step 2: Set-up Partial Fractions

In order to decompose the original fraction, set up a series of fractions: one for the first occurrence of the repeated quadratic factor \(q(x)\), and another for each subsequent occurrence up to the power n of the repeated factor. This will appear as \(A/q(x) + B/q(x)^2 + C/q(x)^3 + ... + Z/q(x)^n\). Note that each fraction's numerator is a variable that we will need to solve for.

Step 3: Match Terms with Original Fraction

Now, equate the set-up fraction series to the original fraction. Multiply each side of the equation by the denominator of the original fraction to clear the denominators.

Step 4: Determine Coefficients

From the equation obtained in step 3, you should have an equation in terms of x. Here, assign different suitable values to x to make the equation easier to solve. Next, solve the series of equations to find the values for A, B, C, ..., Z. These values are the coefficients for each term in the decomposition.

Step 5: Write Out the Partial Fraction Decomposition

Once you have all the values for A, B, C, ..., Z, rewrite the decomposition, replacing the variables with their solved values. This is the partial fraction decomposition of the original rational expression.