Problem 58 - College Algebra Essentials

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

Partial fraction decomposition can be verified without a graph by combining the decomposed fractions. If they sum up to yield the original complex fraction, then the decomposition is correct.

Step 1: Understanding Partial Fraction Decomposition

Partial fraction decomposition is a method in algebra that involves breaking down complex rational fractions or expressions into simpler ones. It's often used in calculus, differential equations, and integrals.

Step 2: Decomposing the Rational Expression

Let's take a generic rational expression, \(\frac{P(x)}{Q(x)}\), where P(x) and Q(x) are polynomials and the degree of P(x) is less than the degree of Q(x). The goal is to express this complex fraction into a sum of simpler fractions, known as a partial fraction decomposition.

Step 3: Verifying the Decomposition

After decomposing the rational expression, the result can be verified by combining the simpler fractions back into the original complex fraction. If the decomposed fractions, when combined, result in the original rational expression, then the decomposition is correct. This can be done by finding common denominators and adding the simpler fractions together.

Step 4: Practice

To verify this conceptually, let's take a look an example: Given the rational expression \(\frac{3x+8}{x^2+5x+6}\), it can be decomposed into \(\frac{1}{x+2} + \frac{2}{x+3}\). To check, we find a common denominator and combine them to get back the original expression. This verifies the accuracy of the decomposition.