IP AMaths Notes (Upper Sec, Year 3-4): 18) Partial Fractions

Study guide08 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Decompose rational expressions with distinct, repeated, and irreducible quadratic factors in IP AMaths.

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Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 18) Partial Fractions cover?
A: Decompose rational expressions with distinct, repeated, and irreducible quadratic factors in IP AMaths.

Partial fractions simplify rational functions for integration and algebraic manipulation.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

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These decomposition templates reflect the SEAB GCE O-Level Additional Mathematics (4049) syllabus for partial fractions (distinct/repeated linear factors and irreducible quadratics).

Status: SEAB O-Level Additional Mathematics 4049 syllabus (exams from 2025) checked 2025-11-30 - scope unchanged; remains the reference for this note.

The core idea is simple: Partial fractions split one rational expression into simpler pieces.

Use it as a working check: Factor the denominator first, choose the correct template, then solve for constants by substitution or coefficient comparison.

Then go one layer deeper: Example: if the denominator is (x + 4)(x - 1), write A/(x + 4) + B/(x - 1), multiply through, then use x = -4 and x = 1.

Choosing the partial fraction template

The denominator decides the template. Factor it fully before writing any constants.

Denominator pattern	Template move	Best way to solve constants
Distinct linear factors, such as $(x + 4)(x - 1)$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB GCE O-Level Additional Mathematics (4049) syllabus (examinations from 2025)

Degree check	First action	Why
Numerator degree is lower	Decompose directly.	The fraction is already proper.
Numerator degree is equal	Divide first.	Partial fractions should be applied only to the proper remainder.
Numerator degree is higher	Divide first.	The quotient is a polynomial part that must stay outside the partial fractions.

Situation after clearing denominators	First move	Why it works	Common trap
A distinct linear factor gives root $x = a$	Substitute $x = a$ .	Every term with $(x - a)$ becomes zero, so one constant is isolated.	Substituting before clearing denominators, which can create division by zero.
A repeated factor such as $(x - a)^2$ appears	Substitute $x = a$ first.	The highest-power term usually survives and gives one constant quickly.	Expecting the same substitution to find every constant in the repeated block.
Constants remain after useful substitutions	Compare coefficients of powers of $x$ .	The identity must match for all values of $x$ , so matching coefficients gives the missing equations.	Comparing coefficients before expanding and collecting like terms cleanly.
The numerator degree is too high	Divide first, then decompose the proper remainder.	Partial fractions apply to the proper fraction part.	Trying to hide the quotient inside the constants.

Denominator factor	Required slots	Why
$(x - a)$	$\dfrac{A}{x - a}$	One linear factor needs one constant numerator.
$(x - a)^2$	$\dfrac{A}{x - a} + \dfrac{B}{(x - a)^2}$
$(x - a)^3$	$\dfrac{A}{x - a} + \dfrac{B}{(x - a)^2} + \dfrac{C}{(x - a)^3}$

Denominator factor	Correct numerator slot	Why	Common trap
$x^2 + 4$	$Bx + C$	A quadratic factor needs enough freedom to match both the $x$ term and constant term after clearing denominators.	Writing only $B$ over the quadratic.
$x^2 + px + q$ that cannot be factorised over real numbers	$Bx + C$	Do not force real linear factors when there are none.	Splitting it into fake factors such as $(x+a)(x+b)$
Linear factor multiplied by irreducible quadratic	$\dfrac{A}{\text{linear}} + \dfrac{Bx+C}{\text{quadratic}}$

IP AMaths Notes (Upper Sec, Year 3-4): 18) Partial Fractions

Choosing the partial fraction template

Sources

Proper-fraction checkpoint

Constant-solving checkpoint

Repeated-factor slot checkpoint

Irreducible-quadratic numerator checkpoint

1 Templates

2 Worked example - Distinct factors

3 Worked example - Repeated factor

4 Worked example - Linear factor with irreducible quadratic

5 Practice Quiz

6 Try this

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