Problem 46 - College Algebra

Question

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

Step-by-Step Solution

Answer

The main idea behind this problem is the concept of 'partial fractions', where a complex fraction is expressed as a sum of simpler fractions. The general steps involve factorizing the denominator, setting up partial fractions, combining like terms, equating and solving to find constants, and finally, representing the original fraction as a sum of simpler partial fractions.

Step 1: Factorization of the Denominator

Start by factorizing the denominator of the rational function. For instance, the denominator \(x^2 - 3x + 2\) can be factored as \((x - 1)(x - 2)\).

Step 2: Setup Partial Fractions

Create a setup for partial fractions. Based on the factors found in Step 1, set up the partial fractions. For example, for our problem \(\frac{8x + 7}{x^2 - 3x + 2}\) will be set up as \(\frac{A}{x - 1} + \frac{B}{x - 2}\), where A and B are constants that we need to discover.

Step 3: Combine Fractions

The next step is to combine these fractions. That would give \(\frac{A(x - 2) + B(x - 1)}{x^2 - 3x + 2}\) which should equal \(\frac{8x + 7}{x^2 - 3x + 2}\)

Step 4: Solve for Constants

As these two rational fractions are equal, their numerators should be equal. That is, \(A(x - 2) + B(x - 1) = 8x + 7\). Now solve for A and B by plugging in appropriate x values (the roots of the denominator). This will let you solve for these constants.

Step 5: Insert Constants into Set-up

Finally, we put these constants into the initial setup from step 2, and obtain the desired partial fractions decomposition.