The Fundamental Theorem of Algebra thus tells us that there are 4 different "simplest'' denominator types:
The method of partial fractions deals with each of them in a slightly different manner.
- linear factors,
- irreducible factors of degree 2,
- repeated linear factors, and
- repeated irreducible factors of degree 2.
- If you encounter a linear factor (x-a) (as (x-4) in
you will use a summand of the form
- An irreducible quadratic term of the form (as in
will contribute two summands of the form
Thus--using different variable names for each term-- the expression above will take the form:
The order in which you write down the terms does not matter, you could have written
- A repeated linear factor will contribute n terms of the form
Last not least, a repeated irreducible factor of degree 2 of the form will contribute 2n terms of the form:
The number of letters you use always equals the degree of the denominator polynomial! This rule does not only apply "globally'', but also "locally'': A repeated linear factor as
yields four expressions with four variable letters.
Predict the form of the right hand side for the following rational
Click on the problem to see the answer!
Click here to continue.
Fri Jul 5 13:54:22 MDT 1996
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