What is a Partial Fraction?Date: 01/08/99 at 06:57:16 From: Axel Ishmael Subject: What is a Partial Fraction: Polynomial? Hello, I was just surfing the Net for math, and I saw this definition in Webster's Dictionary on-line: Main Entry: partial fraction Function: noun Date: 1816 One of the simpler fractions into the sum of which the quotient of two polynomials may be decomposed. What does the above sentence mean? I have a basic understanding of polynomials, but I have never heard of a partial fraction. Thank you. Date: 01/08/99 at 08:23:37 From: Doctor Jerry Subject: Re: What is a Partial Fraction: Polynomial? Hi Axel, For some purposes (such as integration, in calculus) it is useful to "decompose" a fraction like 1/(x^2-a^2). Specifically, notice that if we have: 1/(x^2-a^2) = A/(x-a) + B/(x+a) and if A = 1/(2a) and B = -1/(2a), this equation is satisfied. Thus one can deal more easily with the sum of the simpler "partial fractions" than with 1/(x^2-a^2). There is a theorem in algebra that says that any ratio of polynomials p(x)/q(x) where the degree of p is less than that of q can be decomposed into a sum of partial fractions. To put this theorem into practice, one must factor q into products of linear factors, or powers of linear factors and irreducible quadratic factors, or powers of irreducible quadratic factors (irreducible means no real roots). After all of this, one then must solve for the unknowns (A and B, above, are unknowns). The decomposition of a rational function into partial fractions is computationally labor-intensive. For examples of how to decompose fractions into partial fractions, please see: Partial Fractions http://mathforum.org/dr.math/problems/duke6.8.97.html Partial Fractions http://mathforum.org/dr.math/problems/yanzhen1.29.98.html - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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