Partial Fraction Decompositions

We recall that we can add, subtract, and multiply polynomials together to get another polynomial. But is it possible to divide polynomials and get another polynomial? Unfortunately, we do not get a polynomial all the time. When we have a polynomial divided by another polynomial, for instance, p(x)/q(x), we then have a RATIONAL EXPRESSION in the variable x. Here are a few examples of rational expressions. Note that the sum of two rational expressions is a rational expression and the method for adding them together is the same as adding rational numbers. Therefore we get the identity
    p(x)/s(x) + q(x)/r(x) = [p(x)r(x) + q(x)s(x)]/r(x)s(x)
Which is defined for any x which is not a root of r(x) or s(x). Examples. In all these examples, we are taking two polynomials and adding them together to form a result. What it wanted to take one polynomial and split it into two. This is called the method of PARTIAL FRACTIONS.

There are three different ways of breaking up a rational expression where the numerator is a polynomial of degree 1 or 0 and the denominator is a polynomial of degree 2. Let the rational expression be p(x)/q(x) where p(x) = ax + b and q(x) = cx2 + dx + e.

If q(x) has two roots i and j where i and j are not equal, then

    p(x)/q(x) = [ax+b]/c(x-i)(x-j) = A/c(x-i) + B/(x-j)

If q(x) has one root i, then we get

    p(x)/q(x) = [ax+b]/c(x-i)^2 = B/c(x-i) + B/(x-i)^2

If q(x) has no real roots, then

    p(x)/q(x) = [ax+b]/[cx^2 + dx + e] = ax/[cx^2 + dx + e] + b/[cx^2 + dx + e]

Let's look at a few examples.

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