Problem 56 - Algebra and Trigonometry

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 partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator involves breaking down the rational expression into simpler fractions. The prime factor requires a first-degree polynomial in the numerator of its fraction in the decomposition. The final decomposition obtains by equating coefficients and solving for the unknowns.

Step 1: Identify the Rational Expression

Identify the rational expression which is a fraction where the numerator and the denominator are polynomials. For example, \(\frac{x^2 + 3x + 2}{x^3 - x^2 - x + 1}\).

Step 2: Perform Long Division

If the degree of the numerator is equal to or greater than the degree of the denominator, perform long division or synthetic division to simplify the expression into a polynomial plus a proper fraction where the degree of the numerator is less than the degree of the denominator. This simplification is not required for the given example as the degree of numerator is less than the degree of the denominator.

Step 3: Factorize the Denominator

Factorize the denominator of the resulting proper fraction. For the given example, the denominator \(x^3 - x^2 - x + 1\) doesn't factorize into linear factors but rather into a linear factor and a prime quadratic factor i.e., \((x - 1)(x^2 + 1)\). Prime quadratic factors are those which cannot be factorized further.

Step 4: Write Down General Form of Partial Fractions

Write down the general form of the partial fraction decomposition by assigning a fraction for each factor in the denominator. The numerator of each fraction depends on the degree of the factor in the denominator. For a prime quadratic factor the numerator will be a first degree polynomial. Following these rules, we have: \(\frac{x^2 + 3x + 2}{x^3 - x^2 - x + 1} = \frac{A}{x - 1} + \frac{Bx + C}{x^2+1}\). Here A, B and C are coefficients to be determined.

Step 5: Equating Coefficients and Solving

Multiply out the right side and then collect like terms. You'll get an equation where the coefficients of the powers of x on the left must equal to the right. Write down and solve these 'equations from coefficients' to find the values of A, B and C. Substitute these coefficients back into the partial fractions.