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partial fractions


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Suppose that f(x)/g(x) defines a rational function, so that f(x) and g(x) are polynomials, and suppose that the degree of f(x) is less than the degree of g(x). In general, g(x) can be factorized into a product of some different linear factors, each to some index, and some different irreducible quadratic factors, each to some index. Then the original expression f(x)/g(x) can be written as a sum of terms: corresponding to each (x−α)n in g(x), there are termsand corresponding to each (ax2+bx+c)n in g(x), there are termswhere the real numbers denoted here by capital letters are uniquely determined. The expression f(x)/g(x) is said to have been written in partial fractions. The method, which sounds complicated when stated in general, as above, is more easily understood from examples:The values for the numbers A, B, C,…are found by first multiplying both sides of the equation by the denominator g(x). In the last example, this gives

3x+2=A(x−1)(x2+x+1)+B(x2+x+1)+(Cx+D)(x−1)2.

This has to hold for all values of x, so the coefficients of corresponding powers of x on the two sides can be equated, and this determines the unknowns. In some cases, setting x equal to particular values (in this example, x=1) may determine some of the unknowns more quickly. The method of partial fractions is used in the integration of rational functions.

Subjects: Mathematics.


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