Problem 53
Question
Explain what is meant by the partial fraction decomposition of a rational expression.
Step-by-Step Solution
Verified Answer
The partial fraction decomposition of a rational expression refers to the process of breaking a complex fraction into simpler fractions that are easier to work with.
1Step 1: Introduction to Rational Expression
A rational expression is simply a fraction in which both the numerator and the denominator are polynomials. For example, \(\frac{{3x^2+2x+1}}{{x^2+x-2}}\) is a rational expression.
2Step 2: Partial Fraction Decomposition
Partial fraction decomposition is the process of taking a complex fraction and breaking it down into simpler fractions that are easier to work with. It features prominently in the integration of rational functions.
3Step 3: Example of Partial Fraction Decomposition
For example, consider the rational expression \(\frac{{x^2 + 3x +2}}{{x^3 + x^2 -2x}}\). This can be broken down into \(\frac{1}{x}\) and \(\frac{1}{x-2}\), which are the partial fractions of the original rational expression.
Other exercises in this chapter
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