Partial Fractions

Step 1:

Firstly, we have to check that the fraction is a proper fraction. (Degree of P(x) strictly smaller than Degree of Q(x))

For example,

: Proper fraction

: Improper fraction

: Improper fraction

If the partial fraction is an improper fraction, we need to use long division.

Step 2:

Secondly, we have to see which type of partial fractions it is. There are three main types of partial fractions.

Distinct Linear Factors:

Repeated Linear Factors:

Quadratic Factor:

(Quadratic Factor must be irreducible, i.e. cannot be factorized further)

E.g.

Worked Example

Q: Express in partial fractions.

Step 1: Factorize denominator

Step 2: Use Partial Fractions formula for Distinct Linear Factors

Step 3: Multiply throughout by

Step 4: Choose two suitable values of x.

Let x=3, then 8=B(7), thus B=8/7

Let x=-1/2, then 8=A(-3.5), thus A=-16/7

Step 5: Write down the partial fraction and check using substition (very important to check your answer)

(Ans)

Checking:

Let x=9,

Since both give the same value, the check is a success!