How to do Partial Fractions

Partial Fractions

Step 1:

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

For example,

\displaystyle \frac{3}{2x+9}: Proper fraction

\displaystyle \frac{8x^2}{x^2+3}: Improper fraction

\displaystyle \frac{2x^5+1}{3x^3+x}: 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)



Worked Example

Q: Express \displaystyle\frac{8}{2x^2-5x-3} in partial fractions.

Step 1: Factorize denominator


Step 2: Use Partial Fractions formula for Distinct Linear Factors


Step 3: Multiply throughout by (2x+1)(x-3)


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)

\displaystyle\frac{8}{2x^2-5x-3}=\frac{-16}{7(2x+1)}+\frac{8}{7(x-3)} (Ans)


Let x=9,



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


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