Problem 50 - 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

To verify your result for the partial fraction decomposition, perform the decomposition first, then add the resulting fractions back together. If the original rational expression is regained, it confirms that the decomposition was done correctly.

Step 1: Understanding Partial Fraction Decomposition

Partial fraction decomposition involves expressing a given rational expression as a sum of simpler fractions. It is typically done when we are trying to integrate a complicated rational function or when we are solving certain types of differential equations.

Step 2: Perform the Partial Fraction Decomposition

Carry out the partial fraction decomposition. This could involve factoring the denominator, separating the rational function into a sum of simpler fractions, and then equating coefficients and solving for constants.

Step 3: Recombine the Fractions

After finding the constants, recombine the simpler fractions to form a rational expression.

Step 4: Compare with Original Rational Expression

If the recombined rational expression is identical to the original expression, then it verifies the partial fraction decomposition is correct. This method does not require a graphing utility and purely relies on algebraic operations.