Problem 58 - College Algebra

Question

How can you verify your result for the partial fraction decomposition for a given rational expression without using a graphing utility?

Step-by-Step Solution

Answer

After performing the combination on the partial fractions, we find that the resulting expression is equal to our original expression, hence the result for the partial fraction decomposition is verified to be correct.

Step 1: Write Down the Original Expression

The first step is understanding what original expression we are working with. Let's work with a generic example \( \frac{5x^2+11x+4}{x(x-2)(x+1)} \). That is our original expression.

Step 2: Write Down the Decomposed Form

Next, we will need to know the partial fraction decomposition of our original expression. For examples sake, let's say it's \( \frac{2}{x} - \frac{1}{x-2} + \frac{3}{x+1} \).

Step 3: Perform Combination on Partial Fractions

Now, calculate LCM of the decomposed fractions which is \(x(x-2)(x+1)\). Multiply numerators with the missing terms in their respective denominators and simplify the fractions: \( \left(\frac{2(x-2)(x+1)}{x(x-2)(x+1)}\right) - \left(\frac{x(1)(x+1)}{x(x-2)(x+1)}\right) + \left(\frac{3x(x-2)}{x(x-2)(x+1)}\right) = \frac{5x^2+11x+4}{x(x-2)(x+1)} \)

Step 4: Compare the Expressions

The last step is to compare the final expression after composition with the original expression. If they are equal, then the decomposed form has been correctly verified