Problem 55 - Algebra and Trigonometry

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

In summary, to find the partial fraction decomposition of a rational expression with a repeated linear factor in the denominator, first factorize the denominator, then set up the partial fractions according to the factors found, and finally solve the formed system of linear equations to obtain the values of the coefficients of the partial fractions. Following these steps will yield the partial fraction decomposition of the given rational expression.

Step 1: Factorize the Denominator

The first step is to factorize the denominator of the rational expression. Find all the prime factors and take out common factors in case of repeated linear factors. For instance, for the denominator \(4x^2+8x\), the factorized form will be \(4x(x + 2)\).

Step 2: Setup Partial Fractions

Next, set up the partial fractions. The general rule is that a distinct linear factor \(ax+b\) in the denominator yields a partial fraction of the form \(A/(ax+b)\), while a repeated linear factor \(ax+b)^n\) produces a sum of partial fractions \(A_1/(ax+b) + A_2/(ax+b)^2 + ... + A_n/(ax+b)^n\). For our denominator \(4x(x + 2)\), we would set up partial fractions as \(A/x + B/(x+2)\).

Step 3: Solve for Unknowns

After setting up the partial fractions, combine them over a common denominator which will be the same as the original denominator of the rational expression. This should form an equation that equates the numerator of the original rational expression with a polynomial obtained by combining the partial fractions. This equation will have coefficients of the powers of \(x\), which are known constants, equating each other. Solve the resulting system of linear equations to find the values of \(A\) and \(B\).