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Introduce Things that symbolize greed
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Conclusion Things that symbolize greed
Next up, we’re going to explore *partial fractions*, a technique used to integrate rational functions—that is, functions that are a ratio of two polynomials. When you encounter a rational function, partial fractions can be a game-changer. The basic idea is to decompose a complex fraction into a sum of simpler fractions that are easier to integrate. This method is particularly useful when you have a rational function where the denominator can be factored. The first step involves checking the degree of the numerator and the denominator. If the degree of the numerator is greater than or equal to the degree of the denominator, you'll need to perform polynomial long division first. After that, *factor the denominator* completely into linear and irreducible quadratic factors. For each linear factor, (ax + b), you'll create a term in the partial fraction decomposition of the form A/(ax + b). If a linear factor is repeated, like (ax + b)², you’ll have terms A/(ax + b) + B/(ax + b)². For each irreducible quadratic factor, (ax² + bx + c), you'll create a term of the form (Ax + B)/(ax² + bx + c).
