8.3 Partial Fractions

The Partial Fraction Decomposition of P(x)/Q(x): Q(x) Has Repeated Linear Factors

If (ax + b)^n is a factor of the denominator, this means that the linear factor ax + b is repeated n times. When this occurs, the partial fraction decomposition will contain a sum of n fractions for this factor of the denominator. *If the denominator of the given rational expression is not already factored, then begin by factoring the denominator*

When does a Rational Expression have Distinct Linear Factors in the Denominator?

If the denominator of a rational expression has a linear factor of the form ax+b, then the partial fraction decomposition will contain a term of the form A/ax + b *A is the constant and ax+b is the linear factor* Each distinct linear factor in the denominator produces a partial fraction of the form constant over linear factor. *See example in image*

What is a Partial Fraction and Partial Fraction Decomposition?

The two fractions on the right (in the image) are partial fractions. The sum of these fractions is called the partial fraction decomposition of the rational expression the the left-hand side. Partial fraction decompositions can be written for rational expressions of the form P(x)/Q(x), where P and Q have no common factors and the highest power in the numerator is less than the highest power in the denominator.

Partial Fraction Decomposition with Denominators such as x^3 or x^3-x

When a rational expression contains a power of a factor in the denominator, be sure to set up the partial fraction decomposition to allow for every natural-number power of that factor less than or equal to the power.