Problem 56 - College Algebra Essentials

Question

Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.

Step-by-Step Solution

Answer

The decomposition of a rational expression with a prime quadratic factor in the denominator requires identification of the prime quadratic term, rewriting the expression as a sum of partial fractions, forming a polynomial equation by clearing fractions, comparing coefficients on both sides and solving the system of equations, finally substituting the computed values back to the partial fraction decomposition.

Step 1: Identify the Quadratic Factor

Given a rational expression, identify the prime quadratic factor in the denominator. A prime quadratic factor is a quadratic expression that cannot be factored any further.

Step 2: Write the Original Expression as Equivalent Sum of Partial Fractions

Write the given rational expression equivalent to the partial fractions sum. The sum will be of the form \( A/(linear term) + B/(prime quadratic term) \) where A and B are numbers to be found. For the quadratic term, denominator factor has degree 2, so the numerator must be linear.

Step 3: Clear the Fractions to Form a Polynomial Equation

Multiply each term of the equation by the common denominator of the fractions to clear the fractions and form a polynomial equation.

Step 4: Compare the Coefficients

Match the coefficients of similar terms on both sides of the equation. This now becomes a system of linear equations.

Step 5: Solve the System of Equations

Solve the system of linear equations to find the values of A and B. These values will be used in constructing the final partial fractions.

Step 6: Write the Partial Fraction Decomposition

Substitute the values of A and B into the sum of partial fractions obtained initially to obtain the final partial fraction decomposition of the given rational expression.