Problem 55 - College Algebra

Question

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

Step-by-Step Solution

Answer

The partial fraction decomposition of a rational expression with a repeated linear factor in the denominator can be discovered by splitting the original expression into fraction parts with the repeated factor raised to increasing powers, then equating this to the original expression to find out what the constants in each fraction should be.

Step 1: Identify the Repeated Linear Factor

Look at the denominator of the rational expression and see which is the linear factor that is repeated. Let's call this factor \(x-a\), where 'a' is a constant.

Step 2: Rewrite the Expression

Rewrite the given expression as a sum of fractions with the repeated factor raised to increasing powers in the denominator. Assuming the repeated factor appears n times, write the expression as \(\frac{A}{x-a} + \frac{B}{(x-a)^2} + ... + \frac{Z}{(x-a)^n}\), where A, B,..., Z are constants to be determined.

Step 3: Equate to Original

Now, equate this summed expression to the original rational expression and subsequently, clear the fractions by multiplying both sides by \((x-a)^n\).

Step 4: Expanding and Collecting Like Terms

Expand both sides of the equation and group like terms together, and match coefficients on both sides.

Step 5: Solve for Constants

Solve the systems of linear equations obtained from the previous step to find the constants A, B,..., Z.