Choosing and solving partial fractions
The denominator determines the decomposition. Write the complete template before solving for any constants.
Choose the numerator shape from each factor
The denominator chooses the template
distinct linear factors
A/(x−a) + B/(x−b)
repeated factor (x−a)²
A/(x−a) + B/(x−a)²
irreducible quadratic
(Bx+C)/(x²+px+q)
- Three distinct linear factors need three constant-numerator terms.
- A repeated factor needs one term for every power up to the repeated power.
- An irreducible quadratic factor needs a linear numerator .
Common mistake
For , writing only misses the term.
Use factor roots, then match what remains
Choose x-values that kill terms
N(x) = A(x+2)² + B(x−1)(x+2) + C(x−1)
set x = 1
B-term = 0 · C-term = 0
set x = −2
A-term = 0 · B-term = 0
roots reveal A and C immediately
Worked example
A repeated linear factor
After multiplying by the denominator:
- gives .
- gives .
- The coefficient gives , so .
Answer
Give an irreducible quadratic a linear numerator
Your turnIndependent check [5 marks]
Express in partial fractions.
Show worked answer
gives . Comparing the and coefficients gives and .
Answer