In this explainer, we will learn how to decompose rational expressions into partial fractions when the denominator has nonrepeated linear factors.
We find that in this case, if with different numbers together with (leading coefficient) , and if we also assume that the degree of is less than , the degree of , then for numbers , which can be found by solving a set of linear equations.
For example, consider The denominator has the form that we are interested in. We expect
The simplest way to find these is to multiply this identity through with , or . This gives the identity by the way the factors cancel each other out.
To determine these constants, we consider what happens if we substitute in this equation. It produces the identity such that . What made this work was the fact that the other two terms on the right-hand side vanished at . Likewise, looking at and respectively gives us such that and . We have found the partial fraction decomposition: What happens if ? In this case, we start with the long division of polynomials and write with quotient polynomial and a remainder polynomial of smaller degree than , such that the above method now applies to .
We summarize as in the following.
Partial Fractions: When 𝑄(𝑥) Splits into Linear Factors, None Repeated
- If has a degree greater than or equal to the degree of , apply the division algorithm to take a polynomial quotient from it. After this, assume .
- If has factors , then there are numbers such that the partial fraction decomposition is
- To find , multiply through by to get the equation with each of the a product of the factors (including the constant one) of other than .
- Substituting , we find that but , which gives the relation that we solve for .
- Repeat the steps above to find .
Example 1: Decomposing Rational Expressions into Partial Fractions
Find and such that .
Here, the degree of is less than the degree of . Multiplying through by this gives the equation Setting gives Setting gives
Our solution is therefore
Example 2: Decomposing Rational Expressions into Partial Fractions
Express in partial fractions.
The denominator has a larger degree than the numerator, so we proceed to find such that
Multiplying through by , we get
Then substituting , and gives equations such that