Problem 53 - Algebra and Trigonometry

Question

Explain what is meant by the partial fraction decomposition of a rational expression.

Step-by-Step Solution

Answer

Partial fraction decomposition of a rational expression involves expressing the complex rational function as a sum of simpler fractions, making it easier to further manipulate or integrate. This is achieved by factoring the denominator and equating the original expression to a sum of fractions, then determining the specific coefficients.

Step 1: Definition of a Rational Expression

A rational expression is any expression which can be written as the quotient of two polynomials. It can be represented as P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) is not equal to zero.

Step 2: Definition of Partial Fractions

Partial fraction decomposition involves expressing the rational function as the sum of simpler parts that can be integrated more easily. This is essentially breaking down a complex fraction into simpler, smaller fractions.

Step 3: Process of Decomposition

Partial fraction decomposition involves a few steps. First, factor the denominator of the rational expression. Then, write the expression as a sum of fractions, each with a denominator of one of the factors. After that, find values for the coefficients in the numerator that will make the two sides of the equation equal to each other. Lastly, simplify each of the simpler fractions (if possible).

Step 4: An Example

For example, the rational expression \( \frac{{x^2 + 3x + 2}}{{x^2 - x - 2}} \) has a factorization on the denominator, which allows us to express the whole expression as a sum of simpler fractions: \( \frac{{x^2 + 3x + 2}}{{x^2 - x - 2}} = \frac{A}{x-2} + \frac{B}{x+1} \) for some values A and B which can be determined by matching coefficients and substituting specific values of x.